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Mathematics LibreTexts

16.E: Vector Calculus (Exercises)

  • Page ID
    3612
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

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    16.1: Vector Fields

    1. The domain of vector field \(\vecs{F}=\vecs{F}(x,y)\) is a set of points \((x,y)\) in a plane, and the range of \(\vecs F\) is a set of what in the plane?

    Answer:
    Vectors

    For exercises 2 - 4, determine whether the statement is true or false.

    2. Vector field \(\vecs{F}=⟨3x^2,1⟩\) is a gradient field for both \(ϕ_1(x,y)=x^3+y\) and \(ϕ_2(x,y)=y+x^3+100.\)

    3. Vector field \(\vecs{F}=\dfrac{⟨y,x⟩}{\sqrt{x^2+y^2}}\) is constant in direction and magnitude on a unit circle.

    Answer:
    False

    4. Vector field \(\vecs{F}=\dfrac{⟨y,x⟩}{\sqrt{x^2+y^2}}\) is neither a radial field nor a rotation field.

     

    For exercises 5 - 13, describe each vector field by drawing some of its vectors.

    5. [T] \(\vecs{F}(x,y)=x\,\hat{\mathbf i}+y\,\hat{\mathbf j}\)

    Answer:

    clipboard_ed15464787a3931076aab028ddc5b12d7.png

    6. [T] \(\vecs{F}(x,y)=−y\,\hat{\mathbf i}+x\,\hat{\mathbf j}\)

    7. [T] \(\vecs{F}(x,y)=x\,\hat{\mathbf i}−y\,\hat{\mathbf j}\)

    Answer:

    clipboard_ec0907b086839ad5c262019c65af7add7.png

    8. [T] \(\vecs{F}(x,y)=\,\hat{\mathbf i}+\,\hat{\mathbf j}\)

    9. [T] \(\vecs{F}(x,y)=2x\,\hat{\mathbf i}+3y\,\hat{\mathbf j}\)

    Answer:

    clipboard_ef331d6cd76b60efd3886d0df3b197637.png

    10. [T] \(\vecs{F}(x,y)=3\,\hat{\mathbf i}+x\,\hat{\mathbf j}\)

    11. [T] \(\vecs{F}(x,y)=y\,\hat{\mathbf i}+\sin x\,\hat{\mathbf j}\)

    Answer:

    clipboard_e90f52faea8df656c5e8538475924d4a3.png

    12. [T] \(\vecs F(x,y,z)=x\,\hat{\mathbf i}+y\,\hat{\mathbf j}+z\,\hat{\mathbf k}\)

    13. [T] \(\vecs F(x,y,z)=2x\,\hat{\mathbf i}−2y\,\hat{\mathbf j}−2z\,\hat{\mathbf k}\)

    Answer:

    clipboard_e1eb143e779ce49ea2a01af87d7a3548d.png

    14. [T] \(\vecs F(x,y,z)=yz\,\hat{\mathbf i}−xz\,\hat{\mathbf j}\)

     

    For exercises 15 - 20, find the gradient vector field of each function \(f\).

    15. \(f(x,y)=x\sin y+\cos y\)

    Answer:
    \(\vecs{F}(x,y)=\sin(y)\,\hat{\mathbf i}+(x\cos y−\sin y)\,\hat{\mathbf j}\)

    16. \(f(x,y,z)=ze^{−xy}\)

    17. \(f(x,y,z)=x^2y+xy+y^2z\)

    Answer:
    \(\vecs F(x,y,z)=(2xy+y)\,\hat{\mathbf i}+(x^2+x+2yz)\,\hat{\mathbf j}+y^2\,\hat{\mathbf k}\)

    18. \(f(x,y)=x^2\sin(5y)\)

    19. \(f(x,y)=\ln(1+x^2+2y^2)\)

    Answer:
    \(\vecs{F}(x,y)=\dfrac{2x}{1+x^2+2y^2}\,\hat{\mathbf i}+\dfrac{4y}{1+x^2+2y^2}\,\hat{\mathbf j}\)

    20. \(f(x,y,z)=x\cos\left(\frac{y}{z}\right)\)

     

    21. What is vector field \(\vecs{F}(x,y)\) with a value at \((x,y)\) that is of unit length and points toward \((1,0)\)?

    Answer:
    \(\vecs{F}(x,y)=\dfrac{(1−x)\,\hat{\mathbf i}−y\,\hat{\mathbf j}}{\sqrt{(1−x)^2+y^2}}\)

     

    For exercises 22 - 24, write formulas for the vector fields with the given properties.

    22. All vectors are parallel to the \(x\)-axis and all vectors on a vertical line have the same magnitude.

    23. All vectors point toward the origin and have constant length.

    Answer:
    \(\vecs{F}(x,y)=\dfrac{(y\,\hat{\mathbf i}−x\,\hat{\mathbf j})}{\sqrt{x^2+y^2}}\)

    24. All vectors are of unit length and are perpendicular to the position vector at that point.

     

    25. Give a formula \(\vecs{F}(x,y)=M(x,y)\,\hat{\mathbf i}+N(x,y)\,\hat{\mathbf j}\) for the vector field in a plane that has the properties that \(\vecs{F}=\vecs 0\) at \((0,0)\) and that at any other point \((a,b), \vecs F\) is tangent to circle \(x^2+y^2=a^2+b^2\) and points in the clockwise direction with magnitude \(\|\vecs F\|=\sqrt{a^2+b^2}\).

    Answer:
    \(\vecs{F}(x,y)=y\,\hat{\mathbf i}−x\,\hat{\mathbf j}\)

    26. Is vector field \(\vecs{F}(x,y)=(P(x,y),Q(x,y))=(\sin x+y)\,\hat{\mathbf i}+(\cos y+x)\,\hat{\mathbf j}\) a gradient field?

    27. Find a formula for vector field \(\vecs{F}(x,y)=M(x,y)\,\hat{\mathbf i}+N(x,y)\,\hat{\mathbf j}\) given the fact that for all points \((x,y)\), \(\vecs F\) points toward the origin and \(\|\vecs F\|=\dfrac{10}{x^2+y^2}\).

    Answer:
    \(\vecs{F}(x,y)=\dfrac{−10}{(x^2+y^2)^{3/2}}(x\,\hat{\mathbf i}+y\,\hat{\mathbf j})\)

     

    For exercises 28 - 29, assume that an electric field in the \(xy\)-plane caused by an infinite line of charge along the \(x\)-axis is a gradient field with potential function \(V(x,y)=c\ln\left(\frac{r_0}{\sqrt{x^2+y^2}}\right)\), where \(c>0\) is a constant and \(r_0\) is a reference distance at which the potential is assumed to be zero.

    28. Find the components of the electric field in the \(x\)- and \(y\)-directions, where \(\vecs E(x,y)=−\vecs ∇V(x,y).\)

    29. Show that the electric field at a point in the \(xy\)-plane is directed outward from the origin and has magnitude \(\|\vecs E\|=\dfrac{c}{r}\), where \(r=\sqrt{x^2+y^2}\).

    Answer:
    \(\|\vecs E\|=\dfrac{c}{|r|^2}r=\dfrac{c}{|r|}\dfrac{r}{|r|}\)

     

    A flow line (or streamline) of a vector field \(\vecs F\) is a curve \(\vecs r(t)\) such that \(d\vecs{r}/dt=\vecs F(\vecs r(t))\). If \(\vecs F\) represents the velocity field of a moving particle, then the flow lines are paths taken by the particle. Therefore, flow lines are tangent to the vector field.

    For exercises 30 and 31, show that the given curve \(\vecs c(t)\) is a flow line of the given velocity vector field \(\vecs F(x,y,z)\).

    30. \(\vecs c(t)=⟨ e^{2t},\ln|t|,\frac{1}{t} ⟩,\,t≠0;\quad \vecs F(x,y,z)=⟨2x,z,−z^2⟩\)

    31. \(\vecs c(t)=⟨ \sin t,\cos t,e^t⟩;\quad \vecs F(x,y,z) =〈y,−x,z〉\)

    Answer:
    \(\vecs c′(t)=⟨ \cos t,−\sin t,e^{−t}⟩=\vecs F(\vecs c(t))\)

     

    For exercises 32 - 34, let \(\vecs{F}=x\,\hat{\mathbf i}+y\,\hat{\mathbf j}\), \(\vecs G=−y\,\hat{\mathbf i}+x\,\hat{\mathbf j}\), and \(\vecs H=x\,\hat{\mathbf i}−y\,\hat{\mathbf j}\).  Match \(\vecs F\), \(\vecs G\), and \(\vecs H\) with their graphs.

    32.

    clipboard_ed92420bb9a2a79bf7e0737bdf653eee9.png

    33.

    clipboard_eb5a596effc35031ee81697fe6f857207.png

    Answer:
    \(\vecs H\)

    34.

    clipboard_ec5488ed5d7cc242a516fe39456391eaa.png

    For exercises 35 - 38, let \(\vecs{F}=x\,\hat{\mathbf i}+y\,\hat{\mathbf j}\), \(\vecs G=−y\,\hat{\mathbf i}+x\,\hat{\mathbf j}\), and \(\vecs H=−x\,\hat{\mathbf i}+y\,\hat{\mathbf j}\). Match the vector fields with their graphs in (I)−(IV).

    1. \(\vecs F+\vecs G\)
    2. \(\vecs F+\vecs H\)
    3. \(\vecs G+\vecs H\)
    4. \(−\vecs F+\vecs G\)

    35.

    clipboard_eddbf3f9c662003d8e6e632e520b53134.png

    Answer:
    d. \(−\vecs F+\vecs G\)

    36.

    clipboard_e75a38125572207465d05c29854873433.png

    37.

    clipboard_e0404b6b216ac76764fdd9b22d61f3e46.png

    Answer:
    a. \(\vecs F+\vecs G\)

    38.

    clipboard_e3db08e2f2eec3aeb28cd303fd451bb14.png

    16.2: Line Integrals

    1. True or False? Line integral \(\displaystyle\int _C f(x,y)\,ds\) is equal to a definite integral if \(C\) is a smooth curve defined on \([a,b]\) and if function \(f\) is continuous on some region that contains curve \(C\).

    Answer:
    True

    2. True or False? Vector functions \(\vecs r_1=t\,\hat{\mathbf i}+t^2\,\hat{\mathbf j}, \quad 0≤t≤1,\) and \(\vecs r_2=(1−t)\,\hat{\mathbf i}+(1−t)^2\,\hat{\mathbf j}, \quad 0≤t≤1\), define the same oriented curve.

    3. True or False? \(\displaystyle\int _{−C}(P\,dx+Q\,dy)=\int _C(P\,dx−Q\,dy)\)

    Answer:
    False

    4. True or False? A piecewise smooth curve \(C\) consists of a finite number of smooth curves that are joined together end to end.

    5. True or False? If \(C\) is given by \(x(t)=t,\quad y(t)=t, \quad 0≤t≤1\), then \(\displaystyle\int _Cxy\,ds=\int ^1_0t^2\,dt.\)

    Answer:
    False

    For the following exercises, use a computer algebra system (CAS) to evaluate the line integrals over the indicated path.

    6. [T] \(\displaystyle\int _C(x+y)\,ds\)

    \(C:x=t,y=(1−t),z=0\) from \((0, 1, 0)\) to \((1, 0, 0)\)

    7. [T] \(\displaystyle \int _C(x−y)ds\)

    \(C:\vecs r(t)=4t\,\hat{\mathbf i}+3t\,\hat{\mathbf j}\) when \(0≤t≤2\)

    Answer:
    \(\displaystyle\int _C(x−y)\,ds=10\)

    8. [T] \(\displaystyle\int _C(x^2+y^2+z^2)\,ds\)

    \(C:\vecs r(t)=\sin t\,\hat{\mathbf i}+\cos t\,\hat{\mathbf j}+8t\,\hat{\mathbf k}\) when \(0≤t≤\dfrac{π}{2}\)

    9. [T] Evaluate \(\displaystyle\int _Cxy^4\,ds\), where \(C\) is the right half of circle \(x^2+y^2=16\) and is traversed in the clockwise direction.

    Answer:
    \(\displaystyle\int _Cxy^4\,ds=\frac{8192}{5}\)

    10. [T] Evaluate \(\displaystyle\int _C4x^3ds\), where C is the line segment from \((−2,−1)\) to \((1, 2)\).

    For the following exercises, find the work done.

    11. Find the work done by vector field \(\vecs F(x,y,z)=x\,\hat{\mathbf i}+3xy\,\hat{\mathbf j}−(x+z)\,\hat{\mathbf k}\) on a particle moving along a line segment that goes from \((1,4,2)\) to \((0,5,1)\).

    Answer:
    \(W=8\) units of work

    12. Find the work done by a person weighing 150 lb walking exactly one revolution up a circular, spiral staircase of radius 3 ft if the person rises 10 ft.

    13. Find the work done by force field \(\vecs F(x,y,z)=−\dfrac{1}{2}x\,\hat{\mathbf i}−\dfrac{1}{2}y\,\hat{\mathbf j}+\dfrac{1}{4}\,\hat{\mathbf k}\) on a particle as it moves along the helix \(\vecs r(t)=\cos t\,\hat{\mathbf i}+\sin t\,\hat{\mathbf j}+t\,\hat{\mathbf k}\) from point \((1,0,0)\) to point \((−1,0,3π)\).

    Answer:
    \(W=\frac{3π}{4}\) units of work

    14. Find the work done by vector field \(\vecs{F}(x,y)=y\,\hat{\mathbf i}+2x\,\hat{\mathbf j}\) in moving an object along path \(C\), which joins points \((1, 0)\) and \((0, 1)\).

    15. Find the work done by force \(\vecs{F}(x,y)=2y\,\hat{\mathbf i}+3x\,\hat{\mathbf j}+(x+y)\,\hat{\mathbf k}\) in moving an object along curve \(\vecs r(t)=\cos(t)\,\hat{\mathbf i}+\sin(t)\,\hat{\mathbf j}+16\,\hat{\mathbf k}\), where \(0≤t≤2π\).

    Answer:
    \(W=π\) units of work

    16. Find the mass of a wire in the shape of a circle of radius 2 centered at (3, 4) with linear mass density \(ρ(x,y)=y^2\).

    For the following exercises, evaluate the line integrals.

    17. Evaluate \(\displaystyle\int_C\vecs F·d\vecs{r}\), where \(\vecs{F}(x,y)=−1\,\hat{\mathbf j}\), and \(C\) is the part of the graph of \(y=12x^3−x\) from \((2,2)\) to \((−2,−2)\).

    Answer:
    \(\displaystyle\int _C\vecs F·d\vecs{r}=4\)

    18. Evaluate \(\displaystyle\int _γ(x^2+y^2+z^2)^{−1}ds\), where \(γ\) is the helix \(x=\cos t,y=\sin t,z=t,\) with \(0≤t≤T.\)

    19. Evaluate \(\displaystyle\int _Cyz\,dx+xz\,dy+xy\,dz\) over the line segment from \((1,1,1) \) to \((3,2,0).\)

    Answer:
    \(\displaystyle\int _Cyz\,dx+xz\,dy+xy\,dz=−1\)

    20. Let \(C\) be the line segment from point (0, 1, 1) to point (2, 2, 3). Evaluate line integral \(\displaystyle\int _Cy\,ds.\)

    21. [T] Use a computer algebra system to evaluate the line integral \(\displaystyle\int _Cy^2\,dx+x\,dy\), where \(C\) is the arc of the parabola \(x=4−y^2\) from \((−5, −3)\) to \((0, 2)\).

    Answer:
    \(\displaystyle\int _C(y^2)\,dx+(x)\,dy=\dfrac{245}{6}\)

    22. [T] Use a computer algebra system to evaluate the line integral \(\displaystyle\int _C (x+3y^2)\,dy\) over the path \(C\) given by \(x=2t,y=10t,\) where \(0≤t≤1.\)

    23. [T] Use a CAS to evaluate line integral \(\displaystyle\int _C xy\,dx+y\,dy\) over path \(C\) given by \(x=2t,y=10t\), where \(0≤t≤1\).

    Answer:
    \(\displaystyle\int _Cxy\,dx+y\,dy=\dfrac{190}{3}\)

    24. Evaluate line integral \(\displaystyle\int _C(2x−y)\,dx+(x+3y)\,dy\), where \(C\) lies along the \(x\)-axis from \(x=0\) to \(x=5\).

    26. [T] Use a CAS to evaluate \(\displaystyle\int _C\dfrac{y}{2x^2−y^2}\,ds\), where \(C\) is defined by the parametric equations \(x=t,y=t\), for \(1≤t≤5.\)

    Answer:
    \(\displaystyle\int _C\frac{y}{2x^2−y^2}\,ds=\sqrt{2}\ln 5\)

    27. [T] Use a CAS to evaluate \(\displaystyle\int _Cxy\,ds\), where \(C\) is defined by the parametric equations \(x=t^2,y=4t\), for \(0≤t≤1.\)

    In the following exercises, find the work done by force field \(\vecs F\) on an object moving along the indicated path.

    28. \(\vecs{F}(x,y)=−x \,\hat{\mathbf i}−2y\,\hat{\mathbf j}\)

    \(C:y=x^3\) from \((0, 0)\) to \((2, 8)\)

    Answer:
    \(W=−66\) units of work

    29. \(\vecs{F}(x,y)=2x\,\hat{\mathbf i}+y\,\hat{\mathbf j}\)

    <\(C\): counterclockwise around the triangle with vertices \((0, 0), (1, 0), \) and \((1, 1)\)

    30. \(\vecs F(x,y,z)=x\,\hat{\mathbf i}+y\,\hat{\mathbf j}−5z\,\hat{\mathbf k}\)

    \(C:\vecs r(t)=2\cos t\,\hat{\mathbf i}+2\sin t\,\hat{\mathbf j}+t\,\hat{\mathbf k},0≤t≤2π\)

    Answer:
    \(W=−10π^2\) units of work

    31. Let \(\vecs F\) be vector field \(\vecs{F}(x,y)=(y^2+2xe^y+1)\,\hat{\mathbf i}+(2xy+x^2e^y+2y)\,\hat{\mathbf j}\). Compute the work of integral \(\displaystyle\int _C\vecs F·d\vecs{r}\), where \(C\) is the path \(\vecs r(t)=\sin t\,\hat{\mathbf i}+\cos t\,\hat{\mathbf j},\quad 0≤t≤\dfrac{π}{2}\).

    32. Compute the work done by force \(\vecs F(x,y,z)=2x\,\hat{\mathbf i}+3y\,\hat{\mathbf j}−z\,\hat{\mathbf k}\) along path \(\vecs r(t)=t\,\hat{\mathbf i}+t^2\,\hat{\mathbf j}+t^3\,\hat{\mathbf k}\),where \(0≤t≤1\).

    Answer:
    \(W=2\) units of work

    33. Evaluate \(\displaystyle\int _C\vecs F·d\vecs{r}\), where \(\vecs{F}(x,y)=\dfrac{1}{x+y}\,\hat{\mathbf i}+\dfrac{1}{x+y}\,\hat{\mathbf j}\) and \(C\) is the segment of the unit circle going counterclockwise from \((1,0)\) to \((0, 1)\).

    34. Force \(\vecs F(x,y,z)=zy\,\hat{\mathbf i}+x\,\hat{\mathbf j}+z^2x\,\hat{\mathbf k}\) acts on a particle that travels from the origin to point \((1, 2, 3)\). Calculate the work done if the particle travels:

    1. along the path \((0,0,0)→(1,0,0)→(1,2,0)→(1,2,3)\) along straight-line segments joining each pair of endpoints;
    2. along the straight line joining the initial and final points.
    3. Is the work the same along the two paths?

      clipboard_e7c787ff46860b19ec57c0669a08914af.png

    Answer:
    a. \(W=11\) units of work;
    b. \(W=11\) units of work;
    c. Yes

    35. Find the work done by vector field \(\vecs F(x,y,z)=x\,\hat{\mathbf i}+3xy\,\hat{\mathbf j}−(x+z)\,\hat{\mathbf k}\) on a particle moving along a line segment that goes from \((1, 4, 2)\) to \((0, 5, 1).\)

    36. How much work is required to move an object in vector field \(\vecs{F}(x,y)=y\,\hat{\mathbf i}+3x\,\hat{\mathbf j}\) along the upper part of ellipse \(\dfrac{x^2}{4}+y^2=1\) from \((2, 0)\) to \((−2,0)\)?

    Answer:
    \(W=2π\) units of work

    37. A vector field is given by \(\vecs{F}(x,y)=(2x+3y)\,\hat{\mathbf i}+(3x+2y)\,\hat{\mathbf j}\). Evaluate the line integral of the field around a circle of unit radius traversed in a clockwise fashion.

    38. Evaluate the line integral of scalar function \(xy\) along parabolic path \(y=x^2\) connecting the origin to point \((1, 1)\).

    Answer:
    \(\displaystyle\int _C f\,ds=\dfrac{25\sqrt{5}+1}{120}\)

    39. Find \(\displaystyle\int _Cy^2\,dx+(xy−x^2)\,dy\) along \(C: y=3x\) from \((0, 0)\) to \((1, 3).\)

    40. Find \(\displaystyle\int _Cy^2\,dx+(xy−x^2)\,dy\) along \(C: y^2=9x\) from \((0, 0)\) to \((1, 3).\)

    Answer:
    \(\displaystyle\int _Cy^2\,dx+(xy−x^2)\,dy=6.15\)

    For the following exercises, use a CAS to evaluate the given line integrals.

    41. [T] Evaluate \(\vecs F(x,y,z)=x^2z\,\hat{\mathbf i}+6y\,\hat{\mathbf j}+yz^2\,\hat{\mathbf k}\), where \(C\) is represented by \(\vecs r(t)=t\,\hat{\mathbf i}+t^2\,\hat{\mathbf j}+\ln t \,\hat{\mathbf k},1≤t≤3\).

    42. [T] Evaluate line integral \(\displaystyle\int _γxe^y\,ds\) where, \(γ\) is the arc of curve \(x=e^y\) from \((1,0)\) to \((e,1)\).

    Answer:
    \(\displaystyle\int _γxe^y\,ds≈7.157\)

    43. [T] Evaluate the integral \(\displaystyle\int _γxy^2\,ds\), where \(γ\) is a triangle with vertices \((0, 1, 2), (1, 0, 3)\), and \((0,−1,0)\).

    44. [T] Evaluate line integral \(\displaystyle\int _γ(y^2−xy)\,dx\), where \(γ\) is curve \(y=\ln x\) from \((1, 0)\) toward \((e,1)\).

    Answer:
    \(\displaystyle\int _γ(y^2−xy)\,dx≈−1.379\)

    45. [T] Evaluate line integral \(\displaystyle\int_γ xy^4\,ds\), where \(γ\) is the right half of circle \(x^2+y^2=16\).

    46. [T] Evaluate \(\int C \vecs F⋅d\vecs{r},\int C \vecs F·d\vecs{r},\) where \(\vecs F(x,y,z)=x^2y\,\mathbf{\hat i}+(x−z)\,\mathbf{\hat j}+xyz\,\mathbf{\hat k}\) and

    \(C: \vecs r(t)=t\,\mathbf{\hat i}+t^2\,\mathbf{\hat j}+2\,\mathbf{\hat k},0≤t≤1\).

    Answer:
    \(\displaystyle\int _C \vecs F⋅d\vecs{r}≈−1.133\) units of work

    47. Evaluate \(\displaystyle\int _C \vecs F⋅d\vecs{r}\), where \(\vecs{F}(x,y)=2x\sin y\,\mathbf{\hat i}+(x^2\cos y−3y^2)\,\mathbf{\hat j}\) and

    \(C\) is any path from \((−1,0)\) to \((5, 1)\).

    48. Find the line integral of \(\vecs F(x,y,z)=12x^2\,\mathbf{\hat i}−5xy\,\mathbf{\hat j}+xz\,\mathbf{\hat k}\) over path \(C\) defined by \(y=x^2, z=x^3\) from point \((0, 0, 0)\) to point \((2, 4, 8)\).

    Answer:
    \(\displaystyle\int _C \vecs F⋅d\vecs{r}≈22.857\) units of work

    49. Find the line integral of \(\displaystyle\int _C(1+x^2y)\,ds\), where \(C\) is ellipse \(\vecs r(t)=2\cos t\,\mathbf{\hat i}+3\sin t\,\mathbf{\hat j}\) from \(0≤t≤π.\)

    For the following exercises, find the flux.

    50. Compute the flux of \(\vecs{F}=x^2\,\mathbf{\hat i}+y\,\mathbf{\hat j}\) across a line segment from \((0, 0)\) to \((1, 2).\)

    Answer:
    \(\text{flux}=−\frac{1}{3}\)

    51. Let \(\vecs{F}=5\,\mathbf{\hat i}\) and let \(C\) be curve \(y=0,\) with \(0≤x≤4\).  Find the flux across \(C\).

    52. Let \(\vecs{F}=5\,\mathbf{\hat j}\) and let \(C\) be curve \(y=0,\) with \(0≤x≤4\).  Find the flux across \(C\).

    Answer:
    \(\text{flux}=-20\)

    53. Let \(\vecs{F}=−y\,\mathbf{\hat i}+x\,\mathbf{\hat j}\) and let \(C: \vecs r(t)=\cos t\,\mathbf{\hat i}+\sin t\,\mathbf{\hat j}\) for \(0≤t≤2π\). Calculate the flux across \(C\).

    54. Let \(\vecs{F}=(x^2+y^3)\,\mathbf{\hat i}+(2xy)\,\mathbf{\hat j}\). Calculate flux \(\vecs F\) orientated counterclockwise across the curve \(C: x^2+y^2=9.\)

    Answer:
    \(\text{flux}=0\)

    Complete the rest of the exercises as stated.

    55. Find the line integral of \(\displaystyle\int _C z^2\,dx+y\,dy+2y\,dz,\) where \(C\) consists of two parts: \(C_1\) and \(C_2.\)  \(C_1\) is the intersection of cylinder \(x^2+y^2=16\) and plane \(z=3\) from \((0, 4, 3)\) to \((−4,0,3).\)  \(C_2\) is a line segment from \((−4,0,3)\) to \((0, 1, 5)\).

    56. A spring is made of a thin wire twisted into the shape of a circular helix \(x=2\cos t,\;y=2\sin t,\;z=t.\) Find the mass of two turns of the spring if the wire has a constant mass density of \(ρ\) grams per cm.

    Answer:
    \(m=4πρ\sqrt{5}\) grams

    57. A thin wire is bent into the shape of a semicircle of radius \(a\). If the linear mass density at point \(P\) is directly proportional to its distance from the line through the endpoints, find the mass of the wire.

    58. An object moves in force field \(\vecs F(x,y,z)=y^2\,\mathbf{\hat i}+2(x+1)y\,\mathbf{\hat j}\) counterclockwise from point \((2, 0)\) along elliptical path \(x^2+4y^2=4\) to \((−2,0)\), and back to point \((2, 0)\) along the \(x\)-axis. How much work is done by the force field on the object?

    Answer:
    \(W=0\) units of work

    59. Find the work done when an object moves in force field \(\vecs F(x,y,z)=2x\,\mathbf{\hat i}−(x+z)\,\mathbf{\hat j}+(y−x)\,\mathbf{\hat k}\) along the path given by \(\vecs r(t)=t^2\,\mathbf{\hat i}+(t^2−t)\,\mathbf{\hat j}+3\,\mathbf{\hat k}, \; 0≤t≤1.\)

    60. If an inverse force field \(\vecs F\) is given by \(\vecs F(x,y,z)=\dfrac{k}{‖r‖^3}r\), where \(k\) is a constant, find the work done by \(\vecs F\) as its point of application moves along the \(x\)-axis from \(A(1,0,0)\) to \(B(2,0,0)\).

    Answer:
    \(W=\frac{k}{2}\) units of work

    61. David and Sandra plan to evaluate line integral \(\displaystyle\int _C\vecs F·d\vecs{r}\) along a path in the \(xy\)-plane from \((0, 0)\) to \((1, 1)\). The force field is \(\vecs{F}(x,y)=(x+2y)\,\mathbf{\hat i}+(−x+y^2)\,\mathbf{\hat j}\). David chooses the path that runs along the \(x\)-axis from \((0, 0)\) to \((1, 0)\) and then runs along the vertical line \(x=1\) from \((1, 0)\) to the final point \((1, 1)\). Sandra chooses the direct path along the diagonal line \(y=x\) from \((0, 0)\) to \((1, 1)\). Whose line integral is larger and by how much?

     

    16.3: Conservative Vector Fields

    1. True or False? If vector field \(\vecs F\) is conservative on the open and connected region \(D\), then line integrals of \(\vecs F\) are path independent on \(D\), regardless of the shape of \(D\).

    Answer:
    True

    2. True or False? Function \(\vecs r(t)=\vecs a+t(\vecs b−\vecs a)\), where \(0≤t≤1\), parameterizes the straight-line segment from \(\vecs a\) to \(\vecs b\).

    Answer:
    True

    3. True or False? Vector field \(\vecs F(x,y,z)=(y\sin z)\,\mathbf{\hat i}+(x\sin z)\,\mathbf{\hat j}+(xy\cos z)\,\mathbf{\hat k}\) is conservative.

    Answer:
    True

    4. True or False? Vector field \(\vecs F(x,y,z)=y\,\mathbf{\hat i}+(x+z)\,\mathbf{\hat j}−y\,\mathbf{\hat k}\) is conservative.

    5. Justify the Fundamental Theorem of Line Integrals for \(\displaystyle \int _C\vecs F·d\vecs r\) in the case when \(\vecs{F}(x,y)=(2x+2y)\,\mathbf{\hat i}+(2x+2y)\,\mathbf{\hat j}\) and \(C\) is a portion of the positively oriented circle \(x^2+y^2=25\) from \((5, 0)\) to \((3, 4).\)

    Answer:
    \(\displaystyle \int _C \vecs F·d\vecs r=24\) units of work

    6. [T] Find \(\displaystyle \int _C\vecs F·d\vecs r,\) where \(\vecs{F}(x,y)=(ye^{xy}+\cos x)\,\mathbf{\hat i}+\left(xe^{xy}+\frac{1}{y^2+1}\right)\,\mathbf{\hat j}\) and \(C\) is a portion of curve \(y=\sin x\) from \(x=0\) to \(x=\frac{π}{2}\).

    7. [T] Evaluate line integral \(\displaystyle \int _C\vecs F·d\vecs r\), where \(\vecs{F}(x,y)=(e^x\sin y−y)\,\mathbf{\hat i}+(e^x\cos y−x−2)\,\mathbf{\hat j}\), and \(C\) is the path given by \(\vecs r(t)=(t^3\sin\frac{πt}{2})\,\mathbf{\hat i}−(\frac{π}{2}\cos(\frac{πt}{2}+\frac{π}{2}))\,\mathbf{\hat j}\) for \(0≤t≤1\).

    CNX_Calc_Figure_16_03_201.jpg

    Answer:
    \(\displaystyle \int _C\vecs F·d\vecs r=\left(e−\frac{3π}{2}\right)\) units of work

    For the following exercises, determine whether the vector field is conservative and, if it is, find the potential function.

    8. \(\vecs{F}(x,y)=2xy^3\,\mathbf{\hat i}+3y^2x^2\,\mathbf{\hat j}\)

    9. \(\vecs{F}(x,y)=(−y+e^x\sin y)\,\mathbf{\hat i}+((x+2)e^x\cos y)\,\mathbf{\hat j}\)

    Answer:
    Not conservative

    10. \(\vecs{F}(x,y)=(e^{2x}\sin y)\,\mathbf{\hat i}+(e^{2x}\cos y)\,\mathbf{\hat j}\)

    11. \(\vecs{F}(x,y)=(6x+5y)\,\mathbf{\hat i}+(5x+4y)\,\mathbf{\hat j}\)

    Answer:
    Conservative, \(f(x,y)=3x^2+5xy+2y^2+k\)

    12. \(\vecs{F}(x,y)=(2x\cos(y)−y\cos(x))\,\mathbf{\hat i}+(−x^2\sin(y)−\sin(x))\,\mathbf{\hat j}\)

    13. \(\vecs{F}(x,y)=(ye^x+\sin(y))\,\mathbf{\hat i}+(e^x+x\cos(y))\,\mathbf{\hat j}\)

    Answer:
    Conservative, \(f(x,y)=ye^x+x\sin(y)+k\)

    For the following exercises, evaluate the line integrals using the Fundamental Theorem of Line Integrals.

    14. \(\displaystyle ∮_C(y\,\mathbf{\hat i}+x\,\mathbf{\hat j})·d\vecs r,\) where \(C\) is any path from \((0, 0)\) to \((2, 4)\)

    15. \(\displaystyle ∮_C(2y\,dx+2x\,dy),\) where \(C\) is the line segment from \((0, 0)\) to \((4, 4)\)

    Answer:
    \(\displaystyle ∮_C(2y\,dx+2x\,dy)=32\) units of work

    16. [T] \(\displaystyle ∮_C\left[\arctan\dfrac{y}{x}−\dfrac{xy}{x^2+y^2}\right]\,dx+\left[\dfrac{x^2}{x^2+y^2}+e^{−y}(1−y)\right]\,dy\), where \(C\) is any smooth curve from \((1, 1)\) to \((−1,2).\)

    17. Find the conservative vector field for the potential function \(f(x,y)=5x^2+3xy+10y^2.\)

    Answer:
    \(\vecs{F}(x,y)=(10x+3y)\,\mathbf{\hat i}+(3x+10y)\,\mathbf{\hat j}\)

    For the following exercises, determine whether the vector field is conservative and, if so, find a potential function.

    18. \(\vecs{F}(x,y)=(12xy)\,\mathbf{\hat i}+6(x^2+y^2)\,\mathbf{\hat j}\)

    19. \(\vecs{F}(x,y)=(e^x\cos y)\,\mathbf{\hat i}+6(e^x\sin y)\,\mathbf{\hat j}\)

    Answer:
    \(\vecs F\) is not conservative.

    20. \(\vecs{F}(x,y)=(2xye^{x^2y})\,\mathbf{\hat i}+6(x^2e^{x^2y})\,\mathbf{\hat j}\)

    21. \(\vecs F(x,y,z)=(ye^z)\,\mathbf{\hat i}+(xe^z)\,\mathbf{\hat j}+(xye^z)\,\mathbf{\hat k}\)

    Answer:
    \(\vecs F\) is conservative and a potential function is \(f(x,y,z)=xye^z+k\).

    22. \(\vecs F(x,y,z)=(\sin y)\,\mathbf{\hat i}−(x\cos y)\,\mathbf{\hat j}+\,\mathbf{\hat k}\)

    23. \(\vecs F(x,y,z)=\left(\frac{1}{y}\right)\,\mathbf{\hat i}+(\frac{x}{y^2})\,\mathbf{\hat j}+(2z−1)\,\mathbf{\hat k}\)

    Answer:
    \(\vecs F\) is conservative and a potential function is \(f(x,y,z)=z+k.\)

    24. \(\vecs F(x,y,z)=3z^2\,\mathbf{\hat i}−\cos y\,\mathbf{\hat j}+2xz\,\mathbf{\hat k}\)

    25. \(\vecs F(x,y,z)=(2xy)\,\mathbf{\hat i}+(x^2+2yz)\,\mathbf{\hat j}+y^2\,\mathbf{\hat k}\)

    Answer:
    \(\vecs F\) is conservative and a potential function is \(f(x,y,z)=x^2y+y^2z+k.\)

    For the following exercises, determine whether the given vector field is conservative and find a potential function.

    26. \(\vecs{F}(x,y)=(e^x\cos y)\,\mathbf{\hat i}+6(e^x\sin y)\,\mathbf{\hat j}\)

    27. \(\vecs{F}(x,y)=(2xye^{x^2y})\,\mathbf{\hat i}+6(x^2e^{x^2y})\,\mathbf{\hat j}\)

    Answer:
    \(\vecs F\) is conservative and a potential function is \(f(x,y)=e^{x^2y}+k\)

    For the following exercises, evaluate the integral using the Fundamental Theorem of Line Integrals.

    28. Evaluate \(\displaystyle \int _C\vecs ∇f·d\vecs r\), where \(f(x,y,z)=\cos(πx)+\sin(πy)−xyz\) and \(C\) is any path that starts at \((1,12,2)\) and ends at \((2,1,−1)\).

    29. [T] Evaluate \(\displaystyle \int _C\vecs ∇f·d\vecs r\), where \(f(x,y)=xy+e^x\) and \(C\) is a straight line from \((0,0)\) to \((2,1)\).

    Answer:
    \(\displaystyle \int _C\vecs F·d\vecs r=\left(e^2+1\right)\) units of work

    30. [T] Evaluate \(\displaystyle \int _C\vecs ∇f·d\vecs r,\) where \(f(x,y)=x^2y−x\) and \(C\) is any path in a plane from (1, 2) to (3, 2).

    31. Evaluate \(\displaystyle \int _C\vecs ∇f·d\vecs r,\) where \(f(x,y,z)=xyz^2−yz\) and \(C\) has initial point \((1, 2, 3)\) and terminal point \((3, 5, 2).\)

    Answer:
    \(\displaystyle \int _C\vecs F·d\vecs r=38\) units of work

    For the following exercises, let \(\vecs{F}(x,y)=2xy^2\,\mathbf{\hat i}+(2yx^2+2y)\,\mathbf{\hat j}\) and \(G(x,y)=(y+x)\,\mathbf{\hat i}+(y−x)\,\mathbf{\hat j}\), and let \(C_1\) be the curve consisting of the circle of radius 2, centered at the origin and oriented counterclockwise, and\(C_2\) be the curve consisting of a line segment from \((0, 0)\) to \((1, 1)\) followed by a line segment from \((1, 1)\) to \((3, 1).\)

    CNX_Calc_Figure_16_03_203.jpg

    32. Calculate the line integral of \(\vecs F\) over \(C_1\).

    33. Calculate the line integral of \(\vecs G\) over \(C_1\).

    Answer:
    \(\displaystyle ∮_{C_1}\vecs G·d\vecs r=−8π\) units of work

    34. Calculate the line integral of \(\vecs F\) over \(C_2\).

    35. Calculate the line integral of \(\vecs G\) over \(C_2\).

    Answer:
    \(\displaystyle ∮_{C_2}\vecs F·d\vecs r=7\) units of work

    36. [T] Let \(\vecs F(x,y,z)=x^2\,\mathbf{\hat i}+z\sin(yz)\,\mathbf{\hat j}+y\sin(yz)\,\mathbf{\hat k}\). Calculate \(\displaystyle ∮_C\vecs F·d\vecs{r}\), where \(C\) is a path from \(A=(0,0,1)\) to \(B=(3,1,2)\).

    37. [T] Find line integral \(\displaystyle ∮_C\vecs F·dr\) of vector field \(\vecs F(x,y,z)=3x^2z\,\mathbf{\hat i}+z^2\,\mathbf{\hat j}+(x^3+2yz)\,\mathbf{\hat k}\) along curve \(C\) parameterized by \(\vecs r(t)=(\frac{\ln t}{\ln 2})\,\mathbf{\hat i}+t^{3/2}\,\mathbf{\hat j}+t\cos(πt),1≤t≤4.\)

    Answer:
    \(\displaystyle \int _C\vecs F·d\vecs r=150\) units of work

    For exercises 38 - 40, show that the following vector fields are conservative.  Then calculate \(\displaystyle \int _C\vecs F·d\vecs r\) for the given curve.

    38. \(\vecs{F}(x,y)=(xy^2+3x^2y)\,\mathbf{\hat i}+(x+y)x^2\,\mathbf{\hat j}\); \(C\) is the curve consisting of line segments from \((1,1)\) to \((0,2)\) to \((3,0).\)

    39. \(\vecs{F}(x,y)=\dfrac{2x}{y^2+1}\,\mathbf{\hat i}−\dfrac{2y(x^2+1)}{(y^2+1)^2}\,\mathbf{\hat j}\); \(C\) is parameterized by \(x=t^3−1,\;y=t^6−t\), for \(0≤t≤1.\)

    Answer:
    \(\displaystyle \int _C\vecs F·d\vecs r=−1\) units of work

    40. [T] \(\vecs{F}(x,y)=[\cos(xy^2)−xy^2\sin(xy^2)]\,\mathbf{\hat i}−2x^2y\sin(xy^2)\,\mathbf{\hat j}\); \(C\) is the curve \(\langle e^t,e^{t+1}\rangle,\) for \(−1≤t≤0\).

     

    41. The mass of Earth is approximately \(6×10^{27}g\) and that of the Sun is 330,000 times as much. The gravitational constant is \(6.7×10^{−8}cm^3/s^2·g\). The distance of Earth from the Sun is about \(1.5×10^{12}cm\). Compute, approximately, the work necessary to increase the distance of Earth from the Sun by \(1\;cm\).

    Answer:
    \(4×10^{31}\) erg

    42. [T] Let \(\vecs{F}(x,y,z)=(e^x\sin y)\,\mathbf{\hat i}+(e^x\cos y)\,\mathbf{\hat j}+z^2\,\mathbf{\hat k}\). Evaluate the integral \(\displaystyle \int _C\vecs F·d\vecs r\), where \(\vecs r(t)=\langle\sqrt{t},t^3,e^{\sqrt{t}}\rangle,\) for \(0≤t≤1.\)

    43. [T] Let \(C:[1,2]→ℝ^2\) be given by \(x=e^{t−1},y=\sin\left(\frac{π}{t}\right)\). Use a computer to compute the integral \(\displaystyle \int _C\vecs F·d\vecs r=\int _C 2x\cos y\,dx−x^2\sin y\,dy\), where \(\vecs{F}(x,y)=(2x\cos y)\,\mathbf{\hat i}−(x^2\sin y)\,\mathbf{\hat j}.\)

    Answer:
    \(\displaystyle \int _C\vecs F·d\vecs s=0.4687\) units of work

    44. [T] Use a computer algebra system to find the mass of a wire that lies along the curve \(\vecs r(t)=(t^2−1)\,\mathbf{\hat j}+2t\,\mathbf{\hat k},\) where \(0≤t≤1\), if the density is given by \(d(t) = \dfrac{3}{2}t\).

    45. Find the circulation and flux of field \(\vecs{F}(x,y)=−y\,\mathbf{\hat i}+x\,\mathbf{\hat j}\) around and across the closed semicircular path that consists of semicircular arch \(\vecs r_1(t)=(a\cos t)\,\mathbf{\hat i}+(a\sin t)\,\mathbf{\hat j},\quad 0≤t≤π\), followed by line segment \(\vecs r_2(t)=t\,\mathbf{\hat i},\quad −a≤t≤a.\)

    CNX_Calc_Figure_16_03_204.jpg

    Answer:
    \(\text{circulation}=πa^2\) and \(\text{flux}=0\)

    46. Compute \(\displaystyle \int _C\cos x\cos y\,dx−\sin x\sin y\,dy,\) where \(\vecs r(t)=\langle t,t^2 \rangle, \quad 0≤t≤1.\)

    47. Complete the proof of the theorem titled THE PATH INDEPENDENCE TEST FOR CONSERVATIVE FIELDS by showing that \(f_y=Q(x,y).\)

     

    16.4: Green’s Theorem

    For the following exercises, evaluate the line integrals by applying Green’s theorem.

    1. \(\displaystyle \int_C 2xy\,dx+(x+y)\,dy\), where \(C\) is the path from \((0, 0)\) to \((1, 1)\) along the graph of \(y=x^3\) and from \((1, 1)\) to \((0, 0)\) along the graph of \(y=x\) oriented in the counterclockwise direction

    2. \(\displaystyle \int_C 2xy\,dx+(x+y)\,dy\), where \(C\) is the boundary of the region lying between the graphs of \(y=0\) and \(y=4−x^2\) oriented in the counterclockwise direction

    Answer:
    \(\displaystyle \int_C2xy\,dx+(x+y)\,dy=\frac{32}{3}\) units of work

    3. \(\displaystyle \int_C 2\arctan\left(\frac{y}{x}\right)\,dx+\ln(x^2+y^2)\,dy\), where \(C\) is defined by \(x=4+2\cos θ,\;y=4\sin θ\) oriented in the counterclockwise direction

    4. \(\displaystyle \int_C \sin x\cos y\,dx+(xy+\cos x\sin y)\,dy\), where \(C\) is the boundary of the region lying between the graphs of \(y=x\) and \(y=\sqrt{x}\) oriented in the counterclockwise direction

    Answer:
    \(\displaystyle \int_C\sin x\cos y\,dx+(xy+\cos x\sin y)\,dy=\frac{1}{12}\) units of work

    5. \(\displaystyle \int_C xy\,dx+(x+y)\,dy\), where \(C\) is the boundary of the region lying between the graphs of \(x^2+y^2=1\) and \(x^2+y^2=9\) oriented in the counterclockwise direction

    6. \(\displaystyle ∮_C (−y\,dx+x\,dy)\), where \(C\) consists of line segment \(C_1\) from \((−1,0)\) to \((1, 0)\), followed by the semicircular arc \(C_2\) from \((1, 0)\) back to \((1, 0)\)

    Answer:
    \(\displaystyle ∮_C (−y\,\,dx+x\,\,dy)=π\) units of work

    For the following exercises, use Green’s theorem.

    7. Let \(C\) be the curve consisting of line segments from \((0, 0)\) to \((1, 1)\) to \((0, 1)\) and back to \((0, 0)\). Find the value of \(\displaystyle \int_C xy\,dx+\sqrt{y^2+1}\,dy\).

    8. Evaluate line integral \(\displaystyle \int_C xe^{−2x}\,dx+(x^4+2x^2y^2)\,dy\), where \(C\) is the boundary of the region between circles \(x^2+y^2=1\) and \(x^2+y^2=4\), and is a positively oriented curve.

    Answer:
    \(\displaystyle \int_C xe^{−2x}\,dx+(x^4+2x^2y^2)\,dy=0\) units of work

    9. Find the counterclockwise circulation of field \(\vecs F(x,y)=xy\,\mathbf{\hat i}+y^2\,\mathbf{\hat j}\) around and over the boundary of the region enclosed by curves \(y=x^2\) and \(y=x\) in the first quadrant and oriented in the counterclockwise direction.

    CNX_Calc_Figure_16_04_201.jpg

    10. Evaluate \(\displaystyle ∮_C y^3\,dx−x^3y^2\,dy\), where \(C\) is the positively oriented circle of radius \(2\) centered at the origin.

    Answer:
    \(\displaystyle ∮_C y^3\,dx−x^3y^2\,dy=−24π\) units of work

    11. Evaluate \(\displaystyle ∮_C y^3\,dx−x^3\,dy\), where \(C\) includes the two circles of radius \(2\) and radius \(1\) centered at the origin, both with positive orientation.

    CNX_Calc_Figure_16_04_202.jpg

    12. Calculate \(\displaystyle ∮_C −x^2y\,dx+xy^2\,dy\), where \(C\) is a circle of radius \(2\) centered at the origin and oriented in the counterclockwise direction.

    Answer:
    \(\displaystyle ∮_C −x^2y\,dx+xy^2\,dy=8π\) units of work

    13. Calculate integral \(\displaystyle ∮_C 2[y+x\sin(y)]\,dx+[x^2\cos(y)−3y^2]\,dy\) along triangle \(C\) with vertices \((0, 0), \,(1, 0)\) and \((1, 1)\), oriented counterclockwise, using Green’s theorem.

    14. Evaluate integral \(\displaystyle ∮_C (x^2+y^2)\,dx+2xy\,dy\), where \(C\) is the curve that follows parabola \(y=x^2\) from \((0,0), \,(2,4),\) then the line from \((2, 4)\) to \((2, 0)\), and finally the line from \((2, 0)\) to \((0, 0)\).

    CNX_Calc_Figure_16_04_203.jpg

    Answer:
    \(\displaystyle ∮_C (x^2+y^2)\,dx+2xy\,dy=0\) units of work

    15. Evaluate line integral \(\displaystyle ∮_C (y−\sin(y)\cos(y))\,dx+2x\sin^2(y)\,dy\), where \(C\) is oriented in a counterclockwise path around the region bounded by \(x=−1, \,x=2, \,y=4−x^2\), and \(y=x−2.\)

    CNX_Calc_Figure_16_04_204.jpg

    For the following exercises, use Green’s theorem to find the area.

    16. Find the area between ellipse \(\frac{x^2}{9}+\frac{y^2}{4}=1\) and circle \(x^2+y^2=25\).

    Answer:
    \(A=19π\;\text{units}^2\)

    17. Find the area of the region enclosed by parametric equation

    \(\vecs p(θ)=(\cos(θ)−\cos^2(θ))\,\mathbf{\hat i}+(\sin(θ)−\cos(θ)\sin(θ))\,\mathbf{\hat j}\) for \(0≤θ≤2π.\)

    CNX_Calc_Figure_16_04_205.jpg

    18. Find the area of the region bounded by hypocycloid \(\vecs r(t)=\cos^3(t)\,\mathbf{\hat i}+\sin^3(t)\,\mathbf{\hat j}\). The curve is parameterized by \(t∈[0,2π].\)

    Answer:
    \(A=\frac{3}{8π}\;\text{units}^2\)

    19. Find the area of a pentagon with vertices \((0,4), \,(4,1), \,(3,0), \,(−1,−1),\) and \((−2,2)\).

    20. Use Green’s theorem to evaluate \(\displaystyle \int_{C^+}(y^2+x^3)\,dx+x^4\,dy\), where \(C^+\) is the perimeter of square \([0,1]×[0,1]\) oriented counterclockwise.

    Answer:
    \(\displaystyle \int_{C^+} (y^2+x^3)\,dx+x^4\,dy=0\)

    21. Use Green’s theorem to prove the area of a disk with radius \(a\) is \(A=πa^2\;\text{units}^2\).

    22. Use Green’s theorem to find the area of one loop of a four-leaf rose \(r=3\sin 2θ\). (Hint: \(x\,dy−y\,dx=r^2\,dθ\)).

    Answer:
    \(A=\frac{9π}{8}\;\text{units}^2\)

    23. Use Green’s theorem to find the area under one arch of the cycloid given by the parametric equations: \(x=t−\sin t,\;y=1−\cos t,\;t≥0.\)

    24. Use Green’s theorem to find the area of the region enclosed by curve

    \(\vecs r(t)=t^2\,\mathbf{\hat i}+\left(\frac{t^3}{3}−t\right)\,\mathbf{\hat j},\) for \(−\sqrt{3}≤t≤\sqrt{3}\).

    CNX_Calc_Figure_16_04_206.jpg

    Answer:
    \(A=\frac{8\sqrt{3}}{5}\;\text{units}^2\)

    25. [T] Evaluate Green’s theorem using a computer algebra system to evaluate the integral \(\displaystyle \int_C xe^y\,dx+e^x\,dy\), where \(C\) is the circle given by \(x^2+y^2=4\) and is oriented in the counterclockwise direction.

    26. Evaluate \(\displaystyle \int_C(x^2y−2xy+y^2)\,ds\), where \(C\) is the boundary of the unit square \(0≤x≤1,\;0≤y≤1\), traversed counterclockwise.

    Answer:
    \(\displaystyle \int_C (x^2y−2xy+y^2)\,ds=3\)

    27. Evaluate \(\displaystyle \int_C \frac{−(y+2)\,dx+(x−1)\,dy}{(x−1)^2+(y+2)^2}\), where \(C\) is any simple closed curve with an interior that does not contain point \((1,−2)\) traversed counterclockwise.

    28. Evaluate \(\displaystyle \int_C \frac{x\,dx+y\,dy}{x^2+y^2}\), where \(C\) is any piecewise, smooth simple closed curve enclosing the origin, traversed counterclockwise.

    Answer:
    \(\displaystyle \int_C \frac{x\,dx+y\,dy}{x^2+y^2}=2π\)

    For the following exercises, use Green’s theorem to calculate the work done by force \(\vecs F\) on a particle that is moving counterclockwise around closed path \(C\).

    29. \(\vecs F(x,y)=xy\,\mathbf{\hat i}+(x+y)\,\mathbf{\hat j}, \quad C:x^2+y^2=4\)

    30. \(\vecs F(x,y)=(x^{3/2}−3y)\,\mathbf{\hat i}+(6x+5\sqrt{y})\,\mathbf{\hat j}, \quad C\): boundary of a triangle with vertices \((0, 0), \,(5, 0),\) and \((0, 5)\)

    Answer:
    \(W=\frac{225}{2}\) units of work

    31. Evaluate \(\displaystyle \int_C (2x^3−y^3)\,dx+(x^3+y^3)\,dy\), where \(C\) is a unit circle oriented in the counterclockwise direction.

    32. A particle starts at point \((−2,0)\), moves along the \(x\)-axis to \((2, 0)\), and then travels along semicircle \(y=\sqrt{4−x^2}\) to the starting point. Use Green’s theorem to find the work done on this particle by force field \(\vecs F(x,y)=x\,\mathbf{\hat i}+(x^3+3xy^2)\,\mathbf{\hat j}\).

    Answer:
    \(W=12π\) units of work

    33. David and Sandra are skating on a frictionless pond in the wind. David skates on the inside, going along a circle of radius \(2\) in a counterclockwise direction. Sandra skates once around a circle of radius \(3\), also in the counterclockwise direction. Suppose the force of the wind at point \((x,y)\) is \(\vecs F(x,y)=(x^2y+10y)\,\mathbf{\hat i}+(x^3+2xy^2)\,\mathbf{\hat j}\). Use Green’s theorem to determine who does more work.

    34. Use Green’s theorem to find the work done by force field \(\vecs F(x,y)=(3y−4x)\,\mathbf{\hat i}+(4x−y)\,\mathbf{\hat j}\) when an object moves once counterclockwise around ellipse \(4x^2+y^2=4.\)

    Answer:
    \(W=2π\) units of work

    35. Use Green’s theorem to evaluate line integral \(\displaystyle ∮_C e^{2x}\sin 2y\,dx+e^{2x}\cos 2y\,dy\), where \(C\) is ellipse \(9(x−1)^2+4(y−3)^2=36\) oriented counterclockwise.

    36. Evaluate line integral \(\displaystyle ∮_C y^2\,dx+x^2\,dy\), where \(C\) is the boundary of a triangle with vertices \((0,0), \,(1,1)\), and \((1,0)\), with the counterclockwise orientation.

    Answer:
    \(\displaystyle ∮_C y^2\,dx+x^2\,dy=\frac{1}{3}\) units of work

    37. Use Green’s theorem to evaluate line integral \(\displaystyle \int_C \vecs h·d\vecs r\) if \(\vecs h(x,y)=e^y\,\mathbf{\hat i}−\sin πx\,\mathbf{\hat j}\), where \(C\) is a triangle with vertices \((1, 0), \,(0, 1),\) and \((−1,0),\) traversed counterclockwise.

    38. Use Green’s theorem to evaluate line integral \(\displaystyle \int_C\sqrt{1+x^3}\,dx+2xy\,dy\) where \(C\) is a triangle with vertices \((0, 0), \,(1, 0),\) and \((1, 3)\) oriented clockwise.

    Answer:
    \(\displaystyle \int_C \sqrt{1+x^3}\,dx+2xy\,dy=3\) units of work

    39. Use Green’s theorem to evaluate line integral \(\displaystyle \int_C x^2y\,dx−xy^2\,dy\) where \(C\) is a circle \(x^2+y^2=4\) oriented counterclockwise.

    40. Use Green’s theorem to evaluate line integral \(\displaystyle \int_C \left(3y−e^{\sin x}\right)\,dx+\left(7x+\sqrt{y^4+1}\right)\,dy\) where \(C\) is circle \(x^2+y^2=9\) oriented in the counterclockwise direction.

    Answer:
    \(\displaystyle \int_C \left(3y−e^{\sin x}\right)\,dx+\left(7x+\sqrt{y^4+1}\right)\,dy=36π\) units of work

    41. Use Green’s theorem to evaluate line integral \(\displaystyle \int_C (3x−5y)\,dx+(x−6y)\,dy\), where \(C\) is ellipse \(\frac{x^2}{4}+y^2=1\) and is oriented in the counterclockwise direction.

    A horizontal oval oriented counterclockwise with vertices at (-2,0), (0,-1), (2,0), and (0,1). The region enclosed is shaded.

    42. Let \(C\) be a triangular closed curve from \((0, 0)\) to \((1, 0)\) to \((1, 1)\) and finally back to \((0, 0).\) Let \(\vecs F(x,y)=4y\,\mathbf{\hat i}+6x^2\,\mathbf{\hat j}.\) Use Green’s theorem to evaluate \(\displaystyle ∮_C\vecs F·d\vecs r.\)

    Answer:
    \(\displaystyle ∮_C\vecs F·d\vecs r=2\) units of work

    43. Use Green’s theorem to evaluate line integral \(\displaystyle ∮_C y\,dx−x\,dy\), where \(C\) is circle \(x^2+y^2=a^2\) oriented in the clockwise direction.

    44. Use Green’s theorem to evaluate line integral \(\displaystyle ∮_C (y+x)\,dx+(x+\sin y)\,dy,\) where \(C\) is any smooth simple closed curve joining the origin to itself oriented in the counterclockwise direction.

    Answer:
    \(\displaystyle ∮_C (y+x)\,dx+(x+\sin y)\,dy=0\) units of work

    45. Use Green’s theorem to evaluate line integral \(\displaystyle ∮_C \left(y−\ln(x^2+y^2)\right)\,dx+\left(2\arctan \frac{y}{x}\right)\,dy,\) where \(C\) is the positively oriented circle \((x−2)^2+(y−3)^2=1.\)

    46. Use Green’s theorem to evaluate \(\displaystyle ∮_C xy\,dx+x^3y^3\,dy,\) where \(C\) is a triangle with vertices \((0, 0), \,(1, 0),\) and \((1, 2)\) with positive orientation.

    Answer:
    \(\displaystyle ∮_C xy\,dx+x^3y^3\,dy=2221\) units of work

    47. Use Green’s theorem to evaluate line integral \(\displaystyle \int_C \sin y\,dx+x\cos y\,dy,\) where \(C\) is ellipse \(x^2+xy+y^2=1\) oriented in the counterclockwise direction.

    48. Let \(\vecs F(x,y)=\left(\cos(x^5)−13y^3\right)\,\mathbf{\hat i}+13x^3\,\mathbf{\hat j}.\) Find the counterclockwise circulation \(\displaystyle ∮_C\vecs F·d\vecs r,\) where \(C\) is a curve consisting of the line segment joining \((−2,0)\) and \((−1,0),\) half circle \(y=\sqrt{1−x^2},\) the line segment joining \((1, 0)\) and \((2, 0),\) and half circle \(y=\sqrt{4−x^2}.\)

    Answer:
    \(\displaystyle ∮_C\vecs F·d\vecs r=15π^4\) units of work

    49. Use Green’s theorem to evaluate line integral \(\displaystyle ∫_C \sin(x^3)\,dx+2ye^{x^2}\,dy,\) where \(C\) is a triangular closed curve that connects the points \((0, 0), \,(2, 2),\) and \((0, 2)\) counterclockwise.

    50. Let \(C\) be the boundary of square \(0≤x≤π,\;0≤y≤π,\) traversed counterclockwise. Use Green’s theorem to find \(\displaystyle ∫_C \sin(x+y)\,dx+\cos(x+y)\,dy.\)

    Answer:
    \(\displaystyle \int_C\sin(x+y)\,dx+\cos(x+y)\,dy=4\) units of work

    51. Use Green’s theorem to evaluate line integral \(\displaystyle ∫_C \vecs F·d\vecs r,\) where \(\vecs F(x,y)=(y^2−x^2)\,\mathbf{\hat i}+(x^2+y^2)\,\mathbf{\hat j},\) and \(C\) is a triangle bounded by \(y=0,\;x=3,\) and \(y=x,\) oriented counterclockwise.

    52. Use Green’s Theorem to evaluate integral \(\displaystyle ∫_C \vecs F·d\vecs r,\) where \(\vecs F(x,y)=(xy^2)\,\mathbf{\hat i}+x\,\mathbf{\hat j},\) and \(C\) is a unit circle oriented in the counterclockwise direction.

    Answer:
    \(\displaystyle ∫_C \vecs F·d\vecs r=π\) units of work

    53. Use Green’s theorem in a plane to evaluate line integral \(\displaystyle ∮_C (xy+y^2)\,dx+x^2\,dy,\) where \(C\) is a closed curve of a region bounded by \(y=x\) and \(y=x^2\) oriented in the counterclockwise direction.

    54. Calculate the outward flux of \(\vecs F(x,y)=−x\,\mathbf{\hat i}+2y\,\mathbf{\hat j}\) over a square with corners \((±1,\,±1),\) where the unit normal is outward pointing and oriented in the counterclockwise direction.

    Answer:
    \(\displaystyle ∮_C\vecs F·\vecs N \,ds=4\)

    55. [T] Let \(C\) be circle \(x^2+y^2=4\) oriented in the counterclockwise direction. Evaluate \(\displaystyle ∮_C \left[\left(3y−e^{\arctan x})\,dx+(7x+\sqrt{y^4+1}\right)\,dy\right]\) using a computer algebra system.

    56. Find the flux of field \(\vecs F(x,y)=−x\,\mathbf{\hat i}+y\,\mathbf{\hat j}\) across \(x^2+y^2=16\) oriented in the counterclockwise direction.

    Answer:
    \(\displaystyle ∮_C \vecs F·\vecs N\,ds=32π\)

    57. Let \(\vecs F=(y^2−x^2)\,\mathbf{\hat i}+(x^2+y^2)\,\mathbf{\hat j},\) and let \(C\) be a triangle bounded by \(y=0, \,x=3,\) and \(y=x\) oriented in the counterclockwise direction. Find the outward flux of \(\vecs F\) through \(C\).

    58. [T] Let \(C\) be unit circle \(x^2+y^2=1\) traversed once counterclockwise. Evaluate \(\displaystyle ∫_C \left[−y^3+\sin(xy)+xy\cos(xy)\right]\,dx+\left[x^3+x^2\cos(xy)\right]\,dy\) by using a computer algebra system.

    Answer:
    \(\displaystyle ∫_C \left[−y^3+\sin(xy)+xy\cos(xy)\right]\,dx+\left[x^3+x^2\cos(xy)\right]\,dy=4.7124\) units of work

    59. [T] Find the outward flux of vector field \(\vecs F(x,y)=xy^2\,\mathbf{\hat i}+x^2y\,\mathbf{\hat j}\) across the boundary of annulus \(R=\big\{(x,y):1≤x^2+y^2≤4\big\}=\big\{(r,θ):1≤r≤2,\,0≤θ≤2π\big\}\) using a computer algebra system.

    60. Consider region \(R\) bounded by parabolas \(y=x^2\) and \(x=y^2.\) Let \(C\) be the boundary of \(R\) oriented counterclockwise. Use Green’s theorem to evaluate \(\displaystyle ∮_C  \left(y+e^{\sqrt{x}}\right)\,dx+\left(2x+\cos(y^2)\right)\,dy.\)

    Answer:
    \(\displaystyle ∮_C  \left(y+e^{\sqrt{x}}\right)\,dx+\left(2x+\cos(y^2)\right)\,dy=13\) units of work

     

    16.5: Divergence and Curl

    For the following exercises, determine whether the statement is True or False.

    1.  If the coordinate functions of \(\vecs F : \mathbb{R}^3 \rightarrow \mathbb{R}^3\) have continuous second partial derivatives, then \(\text{curl} \, (\text{div} \,\vecs F)\) equals zero.

    2.  \(\vecs\nabla \cdot (x \,\mathbf{\hat i}  + y  \,\mathbf{\hat j}  + z \,\mathbf{\hat k} ) = 1\).

    Answer:
    False

    3.  All vector fields of the form \(\vecs F(x,y,z) = f(x)\,\mathbf{\hat i} + g(y)\,\mathbf{\hat j} + h(z)\,\mathbf{\hat k}\) are conservative.

    4.  If \(\text{curl} \, \vecs F = \vecs 0\), then \(\vecs F\) is conservative.

    Answer:
    True

    5.  If \(\vecs F\) is a constant vector field then \(\text{div} \,\vecs F = 0\).

    6.  If \(\vecs F\) is a constant vector field then \(\text{curl} \,\vecs F =\vecs 0\).

    Answer:
    True

    For the following exercises, find the curl of \(\vecs F\).

    7.  \(\vecs F(x,y,z) = xy^2z^4\,\mathbf{\hat i} + (2x^2y + z)\,\mathbf{\hat j} + y^3 z^2\,\mathbf{\hat k}\)

    8.  \(\vecs F(x,y,z) = x^2 z\,\mathbf{\hat i} + y^2 x\,\mathbf{\hat j} + (y + 2z)\,\mathbf{\hat k}\)

    Answer:
    \(\text{curl} \,\vecs F = \,\mathbf{\hat i} + x^2\,\mathbf{\hat j} + y^2\,\mathbf{\hat k}\)

    9.  \(\vecs F(x,y,z) = 3xyz^2\,\mathbf{\hat i} + y^2 \sin z\,\mathbf{\hat j} + xe^{2z}\,\mathbf{\hat k}\)

    10.  \(\vecs F(x,y,z) = x^2 yz\,\mathbf{\hat i} + xy^2 z\,\mathbf{\hat j} + xyz^2\,\mathbf{\hat k}\)

    Answer:
    \(\text{curl} \, \vecs F = (xz^2 - xy^2)\,\mathbf{\hat i} + (x^2 y - yz^2)\,\mathbf{\hat j} + (y^2z - x^2z)\,\mathbf{\hat k}\)

    11.  \(\vecs F(x,y,z) = (x \, \cos y)\,\mathbf{\hat i} + xy^2\,\mathbf{\hat j}\)

    12.  \(\vecs F(x,y,z) = (x - y)\,\mathbf{\hat i} + (y - z)\,\mathbf{\hat j} + (z - x)\,\mathbf{\hat k}\)

    Answer:
    \(\text{curl }\, \vecs F = \,\mathbf{\hat i} + \,\mathbf{\hat j} + \,\mathbf{\hat k}\)

    13.  \(\vecs F(x,y,z) = xyz\,\mathbf{\hat i} + x^2y^2z^2 \,\mathbf{\hat j} + y^2z^3 \,\mathbf{\hat k}\)

    14.  \(\vecs F(x,y,z) = xy\,\mathbf{\hat i} + yz \,\mathbf{\hat j} + xz \,\mathbf{\hat k}\)

    Answer:
    \(\text{curl }\, \vecs F = - y\,\mathbf{\hat i} - z \,\mathbf{\hat j} - x \,\mathbf{\hat k}\)

    15.  \(\vecs F(x,y,z) = x^2\,\mathbf{\hat i} + y^2 \,\mathbf{\hat j} + z^2 \,\mathbf{\hat k}\)

    16.  \(\vecs F(x,y,z) = ax\,\mathbf{\hat i} + by \,\mathbf{\hat j} + c \,\mathbf{\hat k}\) for constants \(a, \,b, \,c\).

    Answer:
    \(\text{curl }\, \vecs F = \vecs 0\)

     

    For the following exercises, find the divergence of \(\vecs F\).

    17.  \(\vecs F(x,y,z) = x^2 z\,\mathbf{\hat i} + y^2 x \,\mathbf{\hat j} + (y + 2z) \,\mathbf{\hat k}\)

    18.  \(\vecs F(x,y,z) = 3xyz^2\,\mathbf{\hat i} + y^2 \sin z \,\mathbf{\hat j} + xe^2 \,\mathbf{\hat k}\)

    Answer:
    \(\text{div}\,\vecs F = 3yz^2 + 2y \, \sin z + 2xe^{2z}\)

    19.  \(\vecs{F}(x,y) = (\sin x)\,\mathbf{\hat i} + (\cos y) \,\mathbf{\hat j}\)

    20.  \(\vecs F(x,y,z) = x^2\,\mathbf{\hat i} + y^2 \,\mathbf{\hat j} + z^2 \,\mathbf{\hat k}\)

    Answer:
    \(\text{div}\,\vecs F = 2(x + y + z)\)

    21.  \(\vecs F(x,y,z) = (x - y)\,\mathbf{\hat i} + (y - z) \,\mathbf{\hat j} + (z - x) \,\mathbf{\hat k}\)

    22.  \(\vecs{F}(x,y) = \dfrac{x}{\sqrt{x^2+y^2}}\,\mathbf{\hat i} + \dfrac{y}{\sqrt{x^2+y^2}}\,\mathbf{\hat j}\)

    Answer:
    \(\text{div}\,\vecs F = \dfrac{1}{\sqrt{x^2+y^2}}\)

    23.  \(\vecs{F}(x,y) = x\,\mathbf{\hat i} - y \,\mathbf{\hat j}\)

    24.  \(\vecs F(x,y,z) = ax\,\mathbf{\hat i} + by \,\mathbf{\hat j} + c \,\mathbf{\hat k}\) for constants \(a, \,b, \,c\).

    Answer:
    \(\text{div}\,\vecs F = a + b\)

    25.  \(\vecs F(x,y,z) = xyz\,\mathbf{\hat i} + x^2y^2z^2\,\mathbf{\hat j} + y^2z^3\,\mathbf{\hat k}\)

    26.  \(\vecs F(x,y,z) = xy\,\mathbf{\hat i} + yz\,\mathbf{\hat j} + xz\,\mathbf{\hat k}\)

    Answer:
    \(\text{div}\,\vecs F = x + y + z\)

     

    For exercises 27 & 28, determine whether each of the given scalar functions is harmonic.

    27.  \(u(x,y,z) = e^{-x} (\cos y - \sin y)\)

    28.  \(w(x,y,z) = (x^2 + y^2 + z^2)^{-1/2}\)

    Answer:
    Harmonic

     

    29.  If \(\vecs F(x,y,z) = 2\,\mathbf{\hat i} + 2x j + 3y k\) and \(\vecs G(x,y,z) = x\,\mathbf{\hat i} - y \,\mathbf{\hat j} + z \,\mathbf{\hat k}\), find \(\text{curl} \, (\vecs F \times \vecs G)\).

    30.  If \(\vecs F(x,y,z) = 2\,\mathbf{\hat i} + 2x j + 3y k\) and \(\vecs G(x,y,z) = x\,\mathbf{\hat i} - y \,\mathbf{\hat j} + z \,\mathbf{\hat k}\), find \(\text{div} \, (\vecs F \times \vecs G)\).

    Answer:
    \(\text{div} \, (\vecs F \times \vecs G) = 2z + 3x\)

    31.  Find \(\text{div} \,\vecs F\), given that \(\vecs F = \vecs \nabla f\), where \(f(x,y,z) = xy^3z^2\).

    32.  Find the divergence of \(\vecs F\) for vector field \(\vecs F(x,y,z) = (y^2 + z^2) (x + y) \,\mathbf{\hat i} + (z^2 + x^2)(y + z) \,\mathbf{\hat j} + (x^2 + y^2)(z + x) \,\mathbf{\hat k}\).

    Answer:
    \(\text{div}\,\vecs F = 2r^2\)

    33.  Find the divergence of \(\vecs F\) for vector field \(\vecs F(x,y,z) = f_1(y,z)\,\mathbf{\hat i} + f_2 (x,z) \,\mathbf{\hat j} + f_3 (x,y) \,\mathbf{\hat k}\).

     

    For exercises 34 - 36, use \(r = |\vecs r|\) and \(\vecs r(x,y,z) = \langle x,y,z\rangle\).

    34.  Find the \(\text{curl} \, \vecs r\)

    Answer:
    \(\text{curl} \, \vecs r = \vecs 0\)

    35.  Find the \(\text{curl}\, \dfrac{\vecs r}{r}\).

    36.  Find the \(\text{curl}\, \dfrac{\vecs r}{r^3}\).

    Answer:
    \(\text{curl}\, \dfrac{\vecs r}{r^3} = \vecs 0\)

     

    37.  Let \(\vecs{F}(x,y) = \dfrac{-y\,\mathbf{\hat i}+x\,\mathbf{\hat j}}{x^2+y^2}\), where \(\vecs F\) is defined on \(\big\{(x,y) \in \mathbb{R} | (x,y) \neq (0,0) \big\}\). Find \(\text{curl}\, \vecs F\).

     

    For the following exercises, use a computer algebra system to find the curl of the given vector fields.

    38. [T]  \(\vecs F(x,y,z) = \arctan \left(\dfrac{x}{y}\right)\,\mathbf{\hat i} + \ln \sqrt{x^2 + y^2} \,\mathbf{\hat j}+ \,\mathbf{\hat k}\)

    Answer:
    \(\text{curl }\, \vecs F = \dfrac{2x}{x^2+y^2}\,\mathbf{\hat k}\)

    39. [T]  \(\vecs F(x,y,z) = \sin (x - y)\,\mathbf{\hat i} + \sin (y - z) \,\mathbf{\hat j} + \sin (z - x) \,\mathbf{\hat k}\)

     

    For the following exercises, find the divergence of \(\vecs F\) at the given point.

    40.  \(\vecs F(x,y,z) = \,\mathbf{\hat i} + \,\mathbf{\hat j} + \,\mathbf{\hat k}\) at \((2, -1, 3)\)

    Answer:
    \(\text{div}\,\vecs F = 0\)

    41.  \(\vecs F(x,y,z) = xyz \,\mathbf{\hat i} + y \,\mathbf{\hat j} + z\,\mathbf{\hat k}\) at \((1, 2, 3)\)

    42.  \(\vecs F(x,y,z) = e^{-xy}\,\mathbf{\hat i} + e^{xz}\,\mathbf{\hat j} + e^{yz}\,\mathbf{\hat k}\) at \((3, 2, 0)\)

    Answer:
    \(\text{div}\,\vecs F = 2 - 2e^{-6}\)

    43.  \(\vecs F(x,y,z) = xyz \,\mathbf{\hat i} + y \,\mathbf{\hat j} + z\,\mathbf{\hat k}\) at \((1, 2, 1)\)

    44.  \(\vecs F(x,y,z) = e^x \sin y \,\mathbf{\hat i} - e^x \cos y\,\mathbf{\hat j} \) at \((0, 0, 3)\)

    Answer:
    \(\text{div}\,\vecs F = 0\)

     

    For exercises 45- 49, find the curl of \(\vecs F\) at the given point.

    45.  \(\vecs F(x,y,z) = \,\mathbf{\hat i} + \,\mathbf{\hat j} + \,\mathbf{\hat k}\) at \((2, -1, 3)\)

    46.  \(\vecs F(x,y,z) = xyz \,\mathbf{\hat i} + y\,\mathbf{\hat j} + z\,\mathbf{\hat k}\) at \((1, 2, 3)\)

    Answer:
    \(\text{curl }\, \vecs F = \mathbf{\hat j} - 3\,\mathbf{\hat k}\)

    47.  \(\vecs F(x,y,z) = e^{-xy}\,\mathbf{\hat i} + e^{xz}\,\mathbf{\hat j} + e^{yz}\,\mathbf{\hat k}\) at \((3, 2, 0)\)

    48.  \(\vecs F(x,y,z) = xyz \,\mathbf{\hat i} + y\,\mathbf{\hat j} + z\,\mathbf{\hat k}\) at \((1, 2, 1)\)

    Answer:
    \(\text{curl }\, \vecs F = 2\,\mathbf{\hat j} - \,\mathbf{\hat k}\)

    49.  \(\vecs F(x,y,z) = e^x \sin y \,\mathbf{\hat i} - e^x \cos y\,\mathbf{\hat j} \) at \((0, 0, 3)\)

     

    50.  Let \(\vecs F(x,y,z) = (3x^2 y + az) \,\mathbf{\hat i} + x^3\,\mathbf{\hat j} + (3x + 3z^2)\,\mathbf{\hat k}\).  For what value of \(a\) is \(\vecs F\) conservative?

    Answer:
    \(a = 3\)

    51.  Given vector field \(\vecs{F}(x,y) = \dfrac{1}{x^2+y^2} \langle -y,x\rangle\) on domain \(D = \dfrac{\mathbb{R}^2}{\{(0,0)\}} = \big\{(x,y) \in \mathbb{R}^2 |(x,y) \neq (0,0) \big\}\), is \(\vecs F\) conservative?

    52.  Given vector field \(\vecs{F}(x,y) = \dfrac{1}{x^2+y^2} \langle x,y\rangle\) on domain \(D = \dfrac{\mathbb{R}^2}{\{(0,0)\}}\), is \(\vecs F\) conservative?

    Answer:
    \(\vecs F\) is conservative.

    53.  Find the work done by force field \(\vecs{F}(x,y) = e^{-y}\,\mathbf{\hat i} - xe^{-y}\,\mathbf{\hat j}\) in moving an object from\(P(0, 1)\) to \(Q(2, 0)\). Is the force field conservative?

    54.  Compute divergence \(\vecs F(x,y,z) = (\sinh x)\,\mathbf{\hat i} + (\cosh y)\,\mathbf{\hat j} - xyz\,\mathbf{\hat k}\).

    Answer:
    \(\text{div}\,\vecs F = \cosh x + \sinh y - xy\)

    55.  Compute \(\text{curl }\, \vecs F = (\sinh x)\,\mathbf{\hat i} + (\cosh y)\,\mathbf{\hat j} - xyz\,\mathbf{\hat k}\).

     

    For the following exercises, consider a rigid body that is rotating about the \(x\)-axis counterclockwise with constant angular velocity \(\vecs \omega = \langle a,b,c \rangle\). If \(P\) is a point in the body located at \(\vecs r = x\,\mathbf{\hat i} + y\,\mathbf{\hat j} + z\,\mathbf{\hat k}\), the velocity at \(P\) is given by vector field \(\vecs F = \vecs \omega \times \vecs r\).

    A three dimensional diagram of an object rotating about the x axis in a counterclockwise manner with constant angular velocity w = <a,b,c>. The object is roughly a sphere with pointed ends on the x axis, which cuts it in half. An arrow r is drawn from (0,0,0) to P(x,y,z) and down from P(x,y,z) to the x axis.

    56.  Express \(\vecs F\) in terms of \(\,\mathbf{\hat i},\;\,\mathbf{\hat j},\) and \(\,\mathbf{\hat k}\) vectors.

    Answer:
    \(\vecs F = (bz - cy)\,\mathbf{\hat i}+(cx - az)\,\mathbf{\hat j} + (ay - bx)\,\mathbf{\hat k}\)

    57.  Find \(\text{div} \, F\).

    58.  Find \(\text{curl} \, F\)

    Answer:
    \(\text{curl }\, \vecs F = 2\vecs\omega\)

     

    In the following exercises, suppose that \(\vecs \nabla \cdot \vecs F = 0\) and \(\vecs \nabla \cdot \vecs G = 0\).

    59.  Does \(\vecs F + \vecs G\) necessarily have zero divergence?

    60.  Does \(\vecs F \times \vecs G\) necessarily have zero divergence?

    Answer:
    \(\vecs F \times \vecs G\) does not have zero divergence.

     

    In the following exercises, suppose a solid object in \(\mathbb{R}^3\) has a temperature distribution given by \(T(x,y,z)\). The heat flow vector field in the object is \(\vecs F = - k \vecs \nabla T\), where \(k > 0\) is a property of the material. The heat flow vector points in the direction opposite to that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector is \(\vecs \nabla \cdot \vecs F = -k \vecs \nabla \cdot \vecs \nabla T = - k \vecs \nabla^2 T\).

    61.  Compute the heat flow vector field.

    62.  Compute the divergence.

    Answer:
    \(\vecs \nabla \cdot \vecs F = -200 k [1 + 2(x^2 + y^2 + z^2)] e^{-x^2+y^2+z^2}\)

    63. [T]  Consider rotational velocity field \(\vecs v = \langle 0,10z, -10y \rangle\). If a paddlewheel is placed in plane \(x + y + z = 1\) with its axis normal to this plane, using a computer algebra system, calculate how fast the paddlewheel spins in revolutions per unit time.

    A three dimensional diagram of a rotational velocity field. The arrows are showing a rotation in a clockwise manner. A paddlewheel is shown in plan x + y + z = 1 with n extended out perpendicular to the plane.

     

    16.6: Surface Integrals

    For the following exercises, determine whether the statements are true or false.

    If surface S is given by \(\{(x,y,z) : \, 0 \leq x \leq 1, \, 0 \leq y \leq 1, \, z = 10 \}\), then \[\iint_S f(x,y,z) \, dS = \int_0^1 \int_0^1 f (x,y,10) \, dx dy.\]

    [Hide Solution]

    True

    If surface S is given by \(\{(x,y,z) : \, 0 \leq x \leq 1, \, 0 \leq y \leq 1, \, z = x \}\), then \[\iint_S f(x,y,z) \, dS = \int_0^1 \int_0^1 f (x,y,x) \, dx dy.\]

    Surface

    \(r = \langle v \, \cos u, \, v \, \sin u, \, v^2 \rangle, \, fpr \, 0 \leq u \leq \pi, \, 0 \leq v \leq 2\) is the same surface \(r = \langle \sqrt{v} \, \cos 2u, \, \sqrt{v} \, \sin 2u, \, v \rangle, \, for \, 0 \leq u \leq \dfrac{\pi}{2}, \, 0 \leq v \leq 4\).

    [Hide Solution]

    True

    Given the standard parameterization of a sphere, normal vectors \(t_u \times t_v\) are outward normal vectors.

    For the following exercises, find parametric descriptions for the following surfaces.

    Plane \(3x - 2y + z = 2\)

    [Hide Solution]

    \(r(u,v) = \langle u, \, v, \, 2 - 3u + 2v \rangle \) for \(-\infty \leq u < \infty\) and \( - \infty \leq v < \infty\).

    Paraboloid \(z = x^2 + y^2\), for \(0 \leq z \leq 9\).

    Plane \(2x - 4y + 3z = 16\)

    [Hide Solution]

    \(r(u,v) = \langle u, \, v, \, \dfrac{1}{3} (16 - 2u + 4v) \rangle \) for \(|u| < \infty\) and \(|v| < \infty\).

    The frustum of cone \(z^2 = x^2 + y^2\), for \(2 \leq z \leq 8\)

    The portion of cylinder \(x^2 + y^2 = 9\) in the first octant, for \(0 \leq z \leq 3\)

    A diagram in three dimensions of a section of a cylinder with radius 3. The center of its circular top is (0,0,3). The section exists for x, y, and z between 0 and 3.

    [Hide Solution]

    \(r(u,v) = \langle 3 \, \cos u, \, 3 \, \sin u, \, v \rangle \) for \(0 \leq u \leq \dfrac{\pi}{2}, \, 0 \leq v \leq 3\)

    A cone with base radius r and height h, where r and h are positive constants

    For the following exercises, use a computer algebra system to approximate the area of the following surfaces using a parametric description of the surface.

    [T] Half cylinder \(\{ (r, \theta, z) : \, r = 4, \, 0 \leq \theta \leq \pi, \, 0 \leq z \leq 7 \}\)

    [Hide Solution]

    \(A = 87.9646\)

    [T] Plane \(z = 10 - z - y\) above square \(|x| \leq 2, \, |y| \leq 2\)

    For the following exercises, let S be the hemisphere \(x^2 + y^2 + z^2 = 4\), with \(z \geq 0\), and evaluate each surface integral, in the counterclockwise direction.

    \[\iint_S z\, dS\]

    [Hide Solution]

    \[\iint_S z \, dS = 8\pi\]

    \[\iint_S (x - 2y) \, dS\]

    \[\iint_S (x^2 + y^2) \, dS\]

    [Hide Solution]

    \[\iint_S (x^2 + y^2) \, dS = 16 \pi\]

    For the following exercises, evaluate \[\int \int_S F \cdot N \, ds\]for vector field F, where N is an outward normal vector to surface S.

    \(F(x,y,z) = xi + 2yj = 3zk\), and S is that part of plane \(15x - 12y + 3z = 6\) that lies above unit square \(0 \leq x \leq 1, \, 0 \leq y \leq 1\).

    \(\vecs{F}(x,y) = xi + yj\), and S is hemisphere \(z = \sqrt{1 - x^2 - y^2}\).

    [Hide Solution]

    \[\iint_S F \cdot N \, dS = \dfrac{4\pi}{3}\]

    \(F(x,y,z) = x^2 i + y^2 j + z^2 k\), and S is the portion of plane \(z = y + 1\) that lies inside cylinder \(x^2 + y^2 = 1\).

    A cylinder and an intersecting plane shown in three-dimensions. S is the portion of the plane z = y + 1 inside the cylinder x^2 + y ^2 = 1.

    For the following exercises, approximate the mass of the homogeneous lamina that has the shape of given surface S. Round to four decimal places.

    [T] S is surface \(z = 4 - x - 2y\), with \(z \geq 0, \, x \geq 0, \, y \geq 0; \, \xi = x.\)

    [Hide Solution]

    \(m \approx 13.0639\)

    [T] S is surface \(z = x^2 + y^2\), with \(z \leq 1; \, \xi = z\).

    [T] S is surface \(x^2 + y^2 + x^2 = 5\), with \(z \geq 1; \, \xi = \theta^2\).

    [Hide Solution]

    \(m \approx 228.5313\)

    Evaluate \[\iint_S (y^2 zi + y^3j + xzk) \cdot dS,\] where S is the surface of cube \(-1 \leq x \leq 1, \, -1 \leq y \leq 1\), and \(0 \leq z \leq 2\) in a counterclockwise direction.

     

    Evaluate surface integral \[\iint_S gdS,\] where \(g(x,y,z) = xz + 2x^2 - 3xy\) and S is the portion of plane \(2x - 3y + z = 6\) that lies over unit square R: \(0 \leq x \leq 1, \, 0 \leq y \leq 1\).

    [Hide Solution]

    \[\iint_S gdS = 3 \sqrt{4}\]

    Evaluate \[\iint_S (x + y + z)dS,\] where S is the surface defined parametrically by \(R(u,v) = (2u + v)i + (u - 2v)j + (u + 3v)k\) for \(0 \leq u \leq 1\), and \(0 \leq v \leq 2\).

    A three-dimensional diagram of the given surface, which appears to be a steeply sloped plane stretching through the (x,y) plane.

    [T] Evaluate \[\iint_S (x - y^2 + z)dS,\] where S is the surface defined parametrically by \(R(u,v) = u^2i + vj + uk\) for \(0 \leq u \leq 1, \, 0 \leq v \leq 1\).

    A three-dimensional diagram of the given surface, which appears to be a curve with edges parallel to the y-axis. It increases in x components and decreases in z components the further it is from the y axis.

    [Hide Solution]

    \[\iint_S (x^2 + y - z) \, dS \approx 0.9617\]

    [T] Evaluate where S is the surface defined by \(R(u,v) = ui - u^2j + vk, \, 0 \leq u \leq 2, \, 0 \leq v \leq 1\) for \(0 \leq u \leq 1, \, 0 \leq v \leq 2\).

    Evaluate \[ \iint_S (x^2 + y^2) \, dS,\] where S is the surface bounded above hemisphere \(z = \sqrt{1 - x^2 - y^2}\), and below by plane \(z = 0\).

    [Hide Solution]

    \[\iint_S (x^2 + y^2) \, dS = \dfrac{4\pi}{3}\]

    Evaluate \[ \iint_S (x^2 + y^2 + z^2) \, dS,\] where S is the portion of plane that lies inside cylinder \(x^2 + y^2 = 1\).

    [T] Evaluate \[\iint_S x^2 z dS,\] where S is the portion of cone \(z^2 = x^2 + y^2\) that lies between planes \(z = 1\) and \(z = 4\).

    A diagram of the given upward opening cone in three dimensions. The cone is cut by planes z=1 and z=4.

    [Hide Solution]

    \[\iint_S x^2 zdS = \dfrac{1023\sqrt{2\pi}}{5}\]

    [T] Evaluate \[\iint_S (xz/y) \, dS,\] where S is the portion of cylinder \(x = y^2\) that lies in the first octant between planes \(z = 0, \, z = 5\), and \(y = 4\).

    A diagram of the given cylinder in three-dimensions. It is cut by the planes z=0, z=5, y=1, and y=4.

    [T] Evaluate \[\iint_S (z + y) \, dS,\] where S is the part of the graph of \( z = \sqrt{1 - x^2}\) in the first octant between the xz-plane and plane \(y = 3\).

    A diagram of the given surface in three dimensions in the first octant between the xz-plane and plane y=3. The given graph of z= the square root of (1-x^2) stretches down in a concave down curve from along (0,y,1) to along (1,y,0). It looks like a portion of a horizontal cylinder with base along the xz-plane and height along the y axis.

    [Hide Solution]

    \[\iint_S (z + y) \, dS \approx 10.1\]

    Evaluate \[\iint_S xyz\, dS\] if S is the part of plane \(z = x + y\) that lies over the triangular region in the xy-plane with vertices (0, 0, 0), (1, 0, 0), and (0, 2, 0).

     

    Find the mass of a lamina of density \(\xi (x,y,z) = z\) in the shape of hemisphere \(z = (a^2 - x^2 - y^2)^{1/2}\).

    [Hide Solution]

    \(m = \pi a^3\)

    Compute \[\int \int_S F \cdot N \, dS,\] where \(F(x,y,z) = xi - 5yj + 4zk\) and N is an outward normal vector S, where S is the union of two squares \(S_1\) : \(x = 0, \, 0 \leq y \leq 1, \, 0 \leq z \leq 1\) and \(S_2 \, : \, x = 0, \, 0 \leq x \leq 1, \, 0 \leq y \leq 1\).

    A diagram in three dimensions. It shows the square formed by the components x=0, 0 <= y <= 1, and 0 <= z <= 1. It also shows the square formed by the components z=1, 0 <= x <= 1, and 0 <= y <= 1.

    Compute \[\int \int_S F \cdot N \, dS,\] where \(F(x,y,z) = xyi + zj + (x + y)k\) and N is an outward normal vector S, where S is the triangular region cut off from plane \(x + y + z = 1\) by the positive coordinate axes.

    [Hide Solution]

    \[\iint_S F \cdot N \, dS = \dfrac{13}{24}\]

    Compute \[\int \int_S F \cdot N \, dS,\] where \(F(x,y,z) = 2yzi + (tan^{-1}xz)j + e^{xy}k\) and N is an outward normal vector S, where S is the surface of sphere \(x^2 + y^2 + z^2 = 1\).

    Compute \[\int \int_S F \cdot N \, dS,\] where \(F(x,y,z) = xyzi + xyzj + xyzk\) and N is an outward normal vector S, where S is the surface of the five faces of the unit cube \(0 \leq x \leq 1, \, 0 \leq y \leq 1, \, 0 \leq z \leq 1\) missing \(z = 0\).

    [Hide Solution]

    \[\iint_S F \cdot N \, dS = \dfrac{3}{4}\]

    For the following exercises, express the surface integral as an iterated double integral by using a projection on S on the yz-plane.

    \[\iint_S xy^2 z^3 \, dS;\] S is the first-octant portion of plane \(2x + 3y + 4z = 12\).

    \[\iint_S (x^2 - 2y + z \, dS;\] S is the portion of the graph of \(4x + y = 8\) bounded by the coordinate planes and plane \(z = 6\).

    [Hide Solution]

    \[\int_0^8 \int_0^6 \left( 4 - 3y + \dfrac{1}{16} y^2 + z \right) \left(\dfrac{1}{4} \sqrt{17} \right) \, dz dy\]

    For the following exercises, express the surface integral as an iterated double integral by using a projection on S on the xz-plane

    \[\iint_S xy^2z^3 \, dS;\] S is the first-octant portion of plane \(2x + 3y + 4z = 12\).

    \[\iint_S (x^2 - 2y + z) \, dS;\] is the portion of the graph of \(4x + y = 8\) bounded by the coordinate planes and plane \(z = 6\).

    [Hide Solution]

    \[\int_0^2 \int_0^6 [x^2 - 2 (8 - 4x) + z] \sqrt{17} \, dz dx\]

    Evaluate surface integral \[\iint_S yz \, dS,\] where S is the first-octant part of plane \(x + y + z = \lambda\), where \(\lambda\) is a positive constant.

     

    Evaluate surface integral \[\iint_S (x^2 z + y^2 z) \, dS,\] where S is hemisphere \(x^2 + y^2 + z^2 = a^2, \, z \geq 0.\)

    [Hide Solution]

    \[\iint_S (x^2 z + y^2 z) \, dS = \dfrac{\pi a^5}{2}\]

    Evaluate surface integral \[\iint_S zdA,\] where S is surface \(z = \sqrt{x^2 + y^2}, \, 0 \leq z \leq 2\).

    Evaluate surface integral \[\iint_S x^2 yz \, dS,\] where S is the part of plane \(z = 1 + 2x + 3y\) that lies above rectangle \(0 \leq x \leq 3\) and \(0 \leq y \leq 2\).

    [Hide Solution]

    \[\iint_S x^2 yz \, dS = 171 \sqrt{14}\]

    Evaluate surface integral \[\iint_S yz \, dS,\] where S is plane \(x + y + z = 1\) that lies in the first octant.

     

    Evaluate surface integral \[\iint_S yz \, dS,\] where S is the part of plane \(z = y + 3\) that lies inside cylinder \(x^2 + y^2 = 1\).

    [Hide Solution]

    \[\iint_S yz \, dS = \dfrac{\sqrt{2}\pi}{4}\]

    For the following exercises, use geometric reasoning to evaluate the given surface integrals.

    \[\iint_S \sqrt{x^2 + y^2 + z^2} \, dS,\] where S is surface \(x^2 + y^2 + z^2 = 4, \, z \geq 0\)

    \[\iint_S (xi + yj) \cdot dS,\] where S is surface \(x^2 + y^2 = 4, \, 1 \leq z \leq 3\), oriented with unit normal vectors pointing outward

    [Hide Solution]

    \[\iint_S (xi + yj) \cdot dS = 16 \pi\]

    \[\iint_S (zk) \cdot dS,\] where S is disc \(x^2 + y^2 \leq 9\) on plane \(z = 4\) oriented with unit normal vectors pointing upward

    A lamina has the shape of a portion of sphere \(x^2 + y^2 + z^2 = a^2\) that lies within cone \(z = \sqrt{x^2 + y^2}\). Let S be the spherical shell centered at the origin with radius a, and let C be the right circular cone with a vertex at the origin and an axis of symmetry that coincides with the z-axis. Determine the mass of the lamina if \(\delta(x,y,z) = x^2 y^2 z\).

    A diagram in three dimensions. A cone opens upward with point at the origin and an asic of symmetry that coincides with the z-axis. The upper half of a hemisphere with center at the origin opens downward and is cut off by the xy-plane.

    [Hide Solution]

    \(m = \dfrac{\pi a^7}{192}\)

    A lamina has the shape of a portion of sphere \(x^2 + y^2 + z^2 = a^2\) that lies within cone \(z = \sqrt{x^2 + y^2}\). Let S be the spherical shell centered at the origin with radius a, and let C be the right circular cone with a vertex at the origin and an axis of symmetry that coincides with the z-axis. Suppose the vertex angle of the cone is \(\phi_0\), with \(0 \leq \phi_0 < \dfrac{\pi}{2}\). Determine the mass of that portion of the shape enclosed in the intersection of S and C. Assume \(\delta(x,y,z) = x^2y^2z.\)

    A diagram in three dimensions. A cone opens upward with point at the origin and an asic of symmetry that coincides with the z-axis. The upper half of a hemisphere with center at the origin opens downward and is cut off by the xy-plane.

    A paper cup has the shape of an inverted right circular cone of height 6 in. and radius of top 3 in. If the cup is full of water weighing \(62.5 \, lb/ft^3\), find the total force exerted by the water on the inside surface of the cup.

    [Hide Solution]

    \(F \approx 4.57 \, lb\)

    For the following exercises, the heat flow vector field for conducting objects i \(F = - k\nabla T\), where \(T(x,y,z)\) is the temperature in the object and \(k > 0\) is a constant that depends on the material. Find the outward flux of F across the following surfaces S for the given temperature distributions and assume \(k = 1\).

    \(T(x,y,z) = 100 e^{-x-y}\); S consists of the faces of cube \(|x| \leq 1, \, |y| \leq 1, \, |z| \leq 1\).

    \(T(x,y,z) = - \ln (x^2 + y^2 + z^2)\); S is sphere \(x^2 + y^2 + z^2 = a^2\).

    [Hide Solution]

    \(8\pi a\)

    For the following exercises, consider the radial fields \(F = \dfrac{\langle x,y,z \rangle}{(x^2+y^2+z^2)^{\dfrac{p}{2}}} = \dfrac{r}{|r|^p}\), where p is a real number. Let S consist of spheres A and B centered at the origin with radii \(0 < a < b\). The total outward flux across S consists of the outward flux across the outer sphere B less the flux into S across inner sphere A.

    A diagram in three dimensions of two spheres, one contained completely inside the other. Their centers are both at the origin. Arrows point in toward the origin from outside both spheres.

    Find the total flux across S with \(p = 0\).

    Show that for \(p = 3\) the flux across S is independent of a and b.

    [Hide Solution]

    The net flux is zero.

    16.7: Stokes’ Theorem

    For the following exercises, without using Stokes’ theorem, calculate directly both the flux of \(curl \, F \cdot N\) over the given surface and the circulation integral around its boundary, assuming all are oriented clockwise.

    \(F(x,y,z) = y^2i + z^2j + x^2k\); S is the first-octant portion of plane \(x + y + z = 1\).

    \(F(x,y,z) = zi + xj + yk\); S is hemisphere \(z = (a^2 - x^2 - y^2)^{1/2}\).

    [Hide Solution]

    \[\iint_S (curl \, F \cdot N) \, dS = \pi a^2\]

    \(F(x,y,z) = y^2i + 2xj + 5k\); S is hemisphere \(z = (4 - x^2 - y^2)^{1/2}\).

    \(F(x,y,z) = zi + 2xj + 3yk\); S is upper hemisphere \(z = \sqrt{9 - x^2 - y^2}\).

    [Hide Solution]

    \[\iint_S (curl \, F \cdot N) \, dS = 18 \pi\]

    \(F(x,y,z) = (x + 2z)i + (y - x)j + (z - y)k\); S is a triangular region with vertices (3, 0, 0), (0, 3/2, 0), and (0, 0, 3).

    \(F(x,y,z) = 2yi + 6zj + 3xk\); S is a portion of paraboloid \(z = 4 - x^2 - y^2\) and is above the xy-plane.

    [Hide Solution]

    \[\iint_S (curl \, F \cdot N) \, dS = -8 \pi\]

    For the following exercises, use Stokes’ theorem to evaluate

    \[\iint_S (curl \, F \cdot N) \, dS\] for the vector fields and surface.

    \(F(x,y,z) = xyi - zj\) and S is the surface of the cube \(0 \leq x \leq 1, \, 0 \leq y \leq 1, \, 0 \leq z \leq 1\), except for the face where \(z = 0\) and using the outward unit normal vector.

    \(F(x,y,z) = xyi + x^2 j + z^2 k\); and C is the intersection of paraboloid \(z = x^2 + y^2\) and plane \(z = y\), and using the outward normal vector.

    \[\iint_S (curl \, F \cdot N) \, dS = 0\]

    \(F(x,y,z) = 4yi + z j + 2y k\); and C is the intersection of sphere \(x^2 + y^2 + z^2 = 4\) with plane \(z = 0\), and using the outward normal vector.

    Use Stokes’ theorem to evaluate \[\int_C [2xy^2z \, dx + 2x^2yz \, dy + (x^2y^2 - 2z) \, dz],\] where C is the curve given by \(x = \cos t, \, y = \sin t, \, 0 \leq t \leq 2\pi\), traversed in the direction of increasing t.

    A vector field in three dimensional space. The arrows are larger the further they are from the x, y plane. The arrows curve up from below the x, y plane and slightly above it. The rest tend to curve down and horizontally. An oval-shaped curve is drawn through the middle of the space.

    [Hide Solution]

    \[\int_C F \cdot dS = 0\]

    [T] Use a computer algebraic system (CAS) and Stokes’ theorem to approximate line integral \[\int_C (y \, dx + z \, dy + x \, dz),\] where C is the intersection of plane \(x + y = 2\) and surface \(x^2 + y^2 + z^2 = 2(x + y)\), traversed counterclockwise viewed from the origin.

    [T] Use a CAS and Stokes’ theorem to approximate line integral \[\int_C (3y\, dx + 2z \, dy - 5x \, dz),\] where C is the intersection of the xy-plane and hemisphere \(z = \sqrt{1 - x^2 - y^2}\), traversed counterclockwise viewed from the top—that is, from the positive z-axis toward the xy-plane.

    [Hide Solution]

    \[\int_C F \cdot dS = - 9.4248\]

    [T] Use a CAS and Stokes’ theorem to approximate line integral \[ \int_C [(1 + y) \, z dx + (1 + z) x dy + (1 + x) y dz],\] where C is a triangle with vertices \((1,0,0), \, (0,1,0)\), and \((0,0,1)\) oriented counterclockwise.

    Use Stokes’ theorem to evaluate \[\iint_S curl \, F \cdot dS,\] where \(F(x,y,z) = e^{xy} cos \, zi + x^2 zj + xyk\), and S is half of sphere \(x = \sqrt{1 - y^2 - z^2}\), oriented out toward the positive x-axis.

    [Hide Solution]

    \[\iint_S F \cdot dS = 0\]

    [T] Use a CAS and Stokes’ theorem to evaluate \[\iint_S (curl \, F \cdot N) \, dS,\] where \(F(x,y,z) = x^2 yi + xy^2 j + z^3 k\) and C is the curve of the intersection of plane \(3x + 2y + z = 6\) and cylinder \(x^2 + y^2 = 4\), oriented clockwise when viewed from above.

    [T] Use a CAS and Stokes’ theorem to evaluate \[\iint_S curl \, F \cdot dS,\] where \(F(x,y,z) = \left( \sin(y + z) - yx^2 - \dfrac{y^3}{3}\right)i + x \, \cos (y + z) j + \cos (2y) \, k\) and S consists of the top and the four sides but not the bottom of the cube with vertices \((\pm 1, \, \pm1, \, \pm1)\), oriented outward.

    [Hide Solution]

    \[\iint_S curl \, F \cdot dS = 2.6667\]

    [T] Use a CAS and Stokes’ theorem to evaluate \[\iint_S curl \, F \cdot dS,\] where \(F(x,y,z) = z^2i + 3xyj + x^3y^3k\) and S is the top part of \(z = 5 - x^2 - y^2\) above plane \(z = 1\) and S is oriented upward.

     

    Use Stokes’ theorem to evaluate \[\iint_S (curl \, F \cdot N) dS,\] where \(F(x,y,z) = z^2i + y^2j + xk\) and S is a triangle with vertices (1, 0, 0), (0, 1, 0) and (0, 0, 1) with counterclockwise orientation.

    [Hide Solution]

    \[\iint_S (curl \, F \cdot N)dS = -\dfrac{1}{6}\]

    Use Stokes’ theorem to evaluate line integral \[\int_C (z \, dx + x \, dy + y \, dz),\] where C is a triangle with vertices (3, 0, 0), (0, 0, 2), and (0, 6, 0) traversed in the given order.

    Use Stokes’ theorem to evaluate \[\int_C \left(\dfrac{1}{2} y^2 \, dx + z \, dy + x \, dz \right),\] where C is the curve of intersection of plane \(x + z = 1\) and ellipsoid \(x^2 + 2y^2 + z^2 = 1\), oriented clockwise from the origin.

    A diagram of an intersecting plane and ellipsoid in three dimensional space. There is an orange curve drawn to show the intersection.

    [Hide Solution]

    \[\int_C \left(\dfrac{1}{2} y^2 \, dx + z \, dy + x \, dz \right) = - \dfrac{\pi}{4}\]

    Use Stokes’ theorem to evaluate \[\iint_S (curl \, F \cdot N) dS,\] where \(F(x,y,z) = xi + y^2j + ze^{xy}k\) and S is the part of surface \(z = 1 - x^2 - 2y^2\) with \(z \geq 0\), oriented counterclockwise.

    Use Stokes’ theorem for vector field \(F(x,y,z) = zi + 3xj + 2zk\) where S is surface \(z = 1 - x^2 - 2y^2, \, z \geq 0\), C is boundary circle \(x^2 + y^2 = 1\), and S is oriented in the positive z-direction.

    [Hide Solution]

    \[\iint_S (curl \, F \cdot N)dS = -3\pi]

    Use Stokes’ theorem for vector field \(F(x,y,z) = - \dfrac{3}{2} y^2 i - 2 xyj + yzk\), where S is that part of the surface of plane \(x + y + z = 1\) contained within triangle C with vertices (1, 0, 0), (0, 1, 0), and (0, 0, 1), traversed counterclockwise as viewed from above.

     

    A certain closed path C in plane \(2x + 2y + z = 1\) is known to project onto unit circle \(x^2 + y^2 = 1\) in the xy-plane. Let c be a constant and let \(R(x,y,z) = xi + yj + zk\). Use Stokes’ theorem to evaluate \[\int_C(ck \times R) \cdot dS.\]

    [Hide Solution]

    \[\int_C (ck \times R) \cdot dS = 2\pi c\]

    Use Stokes’ theorem and let C be the boundary of surface \(z = x^2 + y^2\) with \(0 \leq x \leq 2\) and \(0 \leq y \leq 1\) oriented with upward facing normal. Define

    \(F(x,y,z) = [\sin (x^3) + xz] i + (x - yz)j + \cos (z^4) k\) and evaluate \(\int_C F \cdot dS\).

    Let S be hemisphere \(x^2 + y^2 + z^2 = 4\) with \(z \geq 0\), oriented upward. Let \(F(x,y,z) = x^2 e^{yz}i + y^2 e^{xz} j + z^2 e^{xy}k\) be a vector field. Use Stokes’ theorem to evaluate \[\iint_S curl \, F \cdot dS.\]

    [Hide Solution]

    \[\iint_S curl \, F \cdot dS = 0\]

    Let \(F(x,y,z) = xyi + (e^{z^2} + y)j + (x + y)k\) and let S be the graph of function \(y = \dfrac{x^2}{9} + \dfrac{z^2}{9} - 1\) with \(z \leq 0\) oriented so that the normal vector S has a positive y component. Use Stokes’ theorem to compute integral \[\iint_S curl \, F \cdot dS.\]

    Use Stokes’ theorem to evaluate \[ \oint F \cdot dS,\] where \(F(x,y,z) = yi + zj + xk\) and C is a triangle with vertices (0, 0, 0), (2, 0, 0) and \(0,-2,2)\) oriented counterclockwise when viewed from above.

    [Hide Solution]

    \[ \oint F \cdot dS = -4\]

    Use the surface integral in Stokes’ theorem to calculate the circulation of field F, \(F(x,y,z) = x^2y^3 i + j + zk\) around C, which is the intersection of cylinder \(x^2 + y^2 = 4\) and hemisphere \(x^2 + y^2 + z^2 = 16, \, z \geq 0\), oriented counterclockwise when viewed from above.

    A diagram in three dimensions of a vector field and the intersection of a sylinder and hemisphere. The arrows are horizontal and have negative x components for negative y components and have positive x components for positive y components. The curve of intersection between the hemisphere and cylinder is drawn in blue.

    Use Stokes’ theorem to compute \[\iint_S curl \, F \cdot dS.\] where \(F(x,y,z) = i + xy^2j + xy^2 k\) and S is a part of plane \(y + z = 2\) inside cylinder \(x^2 + y^2 = 1\) and oriented counterclockwise.

    A diagram of a vector field in three dimensional space showing the intersection of a plane and a cylinder. The curve where the plane and cylinder intersect is drawn in blue.

    [Hide Solution]

    \[\iint_S curl \, F \cdot dS = 0\]

    Use Stokes’ theorem to evaluate \[\iint_S curl \, F \cdot dS,\] where \(F(x,y,z) = -y^2 i + xj + z^2k\) and S is the part of plane \(x + y + z = 1\) in the positive octant and oriented counterclockwise \(x \geq 0, \, y \geq 0, \, z \geq 0\).

    Let \(F(x,y,z) = xyi + 2zj - 2yk\) and let C be the intersection of plane \(x + z = 5\) and cylinder \(x^2 + y^2 = 9\), which is oriented counterclockwise when viewed from the top. Compute the line integral of F over C using Stokes’ theorem.

    [Hide Solution]

    \[\iint_S curl \, F \cdot dS = -36 \pi\]

    [T] Use a CAS and let \(F(x,y,z) = xy^2i + (yz - x)j + e^{yxz}k\). Use Stokes’ theorem to compute the surface integral of curl F over surface S with inward orientation consisting of cube \([0,1] \times [0,1] \times [0,1]\) with the right side missing.

     

    Let S be ellipsoid \(\dfrac{x^2}{4} + \dfrac{y^2}{9} + z^2 = 1\) oriented counterclockwise and let F be a vector field with component functions that have continuous partial derivatives.

    [Hide Solution]

    \[\iint_S curl \, F \cdot N = 0\]

    Let S be the part of paraboloid \(z = 9 - x^2 - y^2\) with \(z \geq 0\). Verify Stokes’ theorem for vector field \(F(x,y,z) = 3zi + 4xj + 2yk\).

    [T] Use a CAS and Stokes’ theorem to evaluate \[\oint F \cdot dS,\] if \(F(x,y,z) = (3z - \sin x) i + (x^2 + e^y) j + (y^3 - \cos z) k\), where C is the curve given by \(x = \cos t, \, y = \sin t, \, z = 1; \, 0 \leq t \leq 2\pi\).

    [Hide Solution]

    \[\oint_C F \cdot dr = 0\]

    [T] Use a CAS and Stokes’ theorem to evaluate \(F(x,y,z) = 2yi + e^zj - arctan \, xk\) with S as a portion of paraboloid \(z = 4 - x^2 - y^2\) cut off by the xy-plane oriented counterclockwise.

    [T] Use a CAS to evaluate \[\iint_S curl (F) \cdot dS,\] where \(F(x,y,z) = 2zi + 3xj + 5yk\) and S is the surface parametrically by \(r(r,\theta) = r \, \cos \theta i + r \, \sin \theta j + (4 - r^2) k \, (0 \leq \theta \leq 2\pi, \, 0 \leq r \leq 3)\).

    [Hide Solution]

    \[\iint_S curl (F) \cdot dS = 84.8230\]

    Let S be paraboloid \(z = a (1 - x^2 - y^2)\), for \(z \geq 0\), where \(a > 0\) is a real number. Let \(F(x,y,z) = \langle x - y, \, y + z, \, z - x \rangle\). For what value(s) of a (if any) does \[\iint_S (\nabla \times F) \cdot n \, dS\] have its maximum value?

    For the following application exercises, the goal is to evaluate \[A = \iint_S (\nabla \times F) \cdot n \, dS,\] where \(F = \langle xz, \, -xz, \, xy \rangle\) and S is the upper half of ellipsoid \(x^2 + y^2 + 8z^2 = 1\), where \(z \geq 0\).

    Evaluate a surface integral over a more convenient surface to find the value of A.

    [Hide Solution]

    \[A = \iint_S (\nabla \times F) \cdot n \, dS = 0\]

    Evaluate A using a line integral.

    Take paraboloid \(z = x^2 + y^2\), for \(0 \leq z \leq 4\), and slice it with plane \(y = 0\). Let S be the surface that remains for \(y \geq 0\), including the planar surface in the xz-plane. Let C be the semicircle and line segment that bounded the cap of S in plane \(z = 4\) with counterclockwise orientation. Let \(F = \langle 2z + y, \, 2x + z, \, 2y + x \rangle\). Evaluate \[\iint_S (\nabla \times F) \cdot n \, dS.\]

    A diagram of a vector field in three dimensional space where a paraboloid with vertex at the origin, plane at y=0, and plane at z=4 intersect. The remaining surface is the half of a paraboloid under z=4 and above y=0.

    [Hide Solution]

    \[\iint_S (\nabla \times F) \cdot n \, dS = 2\pi\]

    For the following exercises, let S be the disk enclosed by curve \(C \, : \, r(t) = \langle \cos \varphi \, \cos t, \, \sin t, \, \sin \varphi \, \cos t \rangle\), for \(0 \leq t \leq 2\pi\), where \(0 \leq \varphi \leq \dfrac{\pi}{2}\) is a fixed angle.

    What is the length of C in terms of \(\varphi\)?

    What is the circulation of C of vector field \(F = \langle -y, \, -z, \, x \rangle\) as a function of \(\varphi\)?

    [Hide Solution]

     

    \(C = \pi (\cos \varphi - \sin \varphi)\)

    For what value of \(\varphi\) is the circulation a maximum?

    Circle C in plane \(x + y + z = 8\) has radius 4 and center (2, 3, 3). Evaluate \[\oint_C F \cdot dr\] for \(F = \langle 0, \, -z, \, 2y \rangle\), where C has a counterclockwise orientation when viewed from above.

    [Hide Solution]

    \[\oint_C F \cdot dr = 48 \pi\]

    Velocity field \(v = \langle 0, \, 1 -x^2, \, 0 \rangle \), for \(|x| \leq 1\) and \(|z| \leq 1\), represents a horizontal flow in the y-direction. Compute the curl of v in a clockwise rotation.

    Evaluate integral \[ \iint_S (\nabla \times F) \cdot n \, dS,\] where \(F = - xzi + yzj + xye^z k\) and S is the cap of paraboloid \(z = 5 - x^2 - y^2\) above plane \(z = 3\), and n points in the positive z-direction on S.

    [Hide Solution]

    \[ \iint_S (\nabla \times F) \cdot n = 0\]

    For the following exercises, use Stokes’ theorem to find the circulation of the following vector fields around any smooth, simple closed curve C.

    \(F = \nabla (x \, \sin ye^z)\)

    \(F = \langle y^2z^3, \, z2xyz^3, 3xy^2z^2 \rangle \)

    [Hide Solution]

    0

    16.8: The Divergence Theorem

     

    For the following exercises, use a computer algebraic system (CAS) and the divergence theorem to evaluate surface integral \(\int_S F \cdot n \, ds\) for the given choice of F and the boundary surface S. For each closed surface, assume N is the outward unit normal vector.

    [T] \(F(x,y,z) = xi + yj + zk\); S is the surface of cube \(0 \leq x \leq 1, \, 0 \leq y \leq 1, \, 0 < z \leq 1\).

    [T] \(F(x,y,z) = (\cos yz) i + e^{xz}j + 3z^2 k\); S is the surface of hemisphere \(z = \sqrt{4 - x^2 - y^2}\) together with disk \(x^2 + y^2 \leq 4\) in the xy-plane.

    [Hide Solution]

    \[\int_S F \cdot n \, ds = 75.3982\]

    [T] \(F(x,y,z) = (x^2 + y^2 - \x^2)i + x^2 yj + 3zk; \) S is the surface of the five faces of unit cube \(0 \leq x \leq 1, \, 0 \leq y \leq 1, \, 0 < z \leq 1.\)

    [T] F(x,y,z) = xi + yj + zk; \) S is the surface of paraboloid \(z = x^2 + y^2\) for \(0 \leq z \leq 9\).

    [Hide Solution]

    \[\int_S F \cdot n \, ds = 127.2345\]

    [T] \(F(x,y,z) = x^2i + y^2 j + z^2 k\); S is the surface of sphere \(x^2 + y^2 + z^2 = 4\).

    [T] \(F(x,y,z) = xi + yj + (z^2 - 1)k\); S is the surface of the solid bounded by cylinder \( x^2 + y^2 = 4\) and planes \(z = 0\) and \(z = 1\).

    [Hide Solution]

    \[\iint_S F \cdot n \, ds = 37.699\]

    [T] \(F(x,y,z) = x^2i + y^2 j + z^2 k\); S is the surface bounded above by sphere \(\rho = 2\) and below by cone \(\varphi = \dfrac{\pi}{4}\) in spherical coordinates. (Think of S as the surface of an “ice cream cone.”)

    [T] \(F(x,y,z) = x^3i + y^3 j + 3a^2z k \, (constant \, a > 0)\); S is the surface bounded by cylinder \(x^2 + y^2 = a^2\) and planes \(z = 0\) and \(z = 1\).

    [Hide Solution]

    \[\int_S F \cdot n \, ds = \dfrac{9\pi a^4}{2}\]

    [T] Surface integral \(\iint_S F \cdot dS\), where S is the solid bounded by paraboloid \(z = x^2 + y^2\) and plane \(z = 4\), and \(F(x,y,z) = (x + y^2z^2)i + (y + z^2x^2)j + (z + x^2y^2) k\)

    Use the divergence theorem to calculate surface integral \(\iint_S F \cdot dS\), where \(F(x,y,z) = (e^{y^2} i + (y + \sin (z^2))j + (z - 1)k\) and S is upper hemisphere \(x^2 + y^2 + z^2 = 1, \, z \geq 0\), oriented upward.

    [Hide Solution]

    \[\iint_S F \cdot dS = \dfrac{\pi}{3}\]

    Use the divergence theorem to calculate surface integral \(\iint_S F \cdot dS\), where \(F(x,y,z) = x^4i - x^3z^2j + 4xy^2zk\) and S is the surface bounded by cylinder \(x^2 + y^2 = 1\) and planes \(z = x + 2\) and \(z = 0\).

    Use the divergence theorem to calculate surface integral \(\iint_S F \cdot dS\), when \(F(x,y,z) = x^2z^3 i + 2xyz^3j + xz^4 k\) and S is the surface of the box with vertices \((\pm 1, \, \pm 2, \, \pm 3)\).

    [Hide Solution]

    \[\iint_S F \cdot dS = 0\]

    Use the divergence theorem to calculate surface integral \(\iint_S F \cdot dS\), when \(F(x,y,z) = z \, tan^{-1} (y^2)i + z^3 \ln(x^2 + 1) j + zk\) and S is a part of paraboloid \(x^2 + y^2 + z = 2\) that lies above plane \(z = 1\) and is oriented upward.

    [T] Use a CAS and the divergence theorem to calculate flux \(\iint_S F \cdot dS\), where \(F(x,y,z) = (x^3 + y^3)i + (y^3 + z^3)j + (z^3 + x^3)k \) and S is a sphere with center (0, 0) and radius 2.

    [Hide Solution]

    \[\iint_S F \cdot dS = 241.2743\]

    Use the divergence theorem to compute the value of flux integral (\iint_S F \cdot dS\), where \(F(x,y,z) = (y^3 + 3x)i + (xz + y)j + [z + x^4 \cos (x^2y)]k\) and S is the area of the region bounded by \(x^2 + y^2 = 1, \, x \geq 0, \, y \geq 0\), and \(0 \leq z \leq 1\).

    A vector field in three dimensions, with focus on the area with x > 0, y>0, and z>0. A quarter of a cylinder is drawn with center on the z axis. The arrows have positive x, y, and z components; they point away from the origin.

    Use the divergence theorem to compute flux integral (\iint_S F \cdot dS\), where \(F(x,y,z) = yj - zk\) and S consists of the union of paraboloid \(y = x^2 + z^2, \, 0 \leq y \leq 1\), and disk \(x^2 + z^2 \leq 1, \, y = 1\), oriented outward. What is the flux through just the paraboloid?

    [Hide Solution]

    \[\iint_S F \cdot dS = -\pi\]

    Use the divergence theorem to compute flux integral (\iint_S F \cdot dS\), where \(F(x,y,z) = x + yj + z^4 k\) and S is a part of cone \(z = \sqrt{x^2 + y^2}\) beneath top plane \(z = 1\) oriented downward.

    Use the divergence theorem to calculate surface integral \(\iint_S F \cdot dS\) for \(F(x,y,z) = x^4i - x^3z^2j + 4xy^2 zk\), where S is the surface bounded by cylinder \(x^2 + y^2 = 1\) and planes \(z = x + 2\) and \(z = 0\).

    [Hide Solution]

    \[\iint_S F \cdot dS = \dfrac{2\pi}{3}\]

    Consider \(F(x,y,z) = x^2i + xyj + (z + 1)k\). Let E be the solid enclosed by paraboloid \(z = 4 - x^2 - y^2\) and plane \(z = 0\) with normal vectors pointing outside E. Compute flux F across the boundary of E using the divergence theorem.

     

    For the following exercises, use a CAS along with the divergence theorem to compute the net outward flux for the fields across the given surfaces S.

    [T] \(F = \langle x,\, -2y, \, 3z \rangle; \) S is sphere \(\{(x,y,z) : x^2 + y^2 + z^2 = 6 \}\).

    [Hide Solution]

    \(15\sqrt{6}\pi\)

    [T] \(F = \langle x, \, 2y, \, z \rangle\); S is the boundary of the tetrahedron in the first octant formed by plane \(x + y + z = 1\).

    [T[ \(F = \langle y - 2x, \, x^3 - y, \, y^2 - z \rangle\); S is sphere \(\{(x,y,z) : x^2 + y^2 + z^2 = 4\}.\)

    [Hide Solution]

    \(-\dfrac{128}{3} \pi\)

    [T] \(F = \langle x,y,z \rangle\); S is the surface of paraboloid \(z = 4 - x^2 - y^2\), for \(z \geq 0\), plus its base in the xy-plane.

    For the following exercises, use a CAS and the divergence theorem to compute the net outward flux for the vector fields across the boundary of the given regions D.

    [T] \(F = \langle z - x, \, x - y, \, 2y - z \rangle\); D is the region between spheres of radius 2 and 4 centered at the origin.

    [Hide Solution]

    \(-703.7168\)

    [T] \(F = \dfrac{r}{|r|} = \dfrac{\langle x,y,z\rangle}{\sqrt{x^2+y^2+z^2}}\); D is the region between spheres of radius 1 and 2 centered at the origin.

    {T} \(F = \langle x^2, \, -y^2, \, z^2 \rangle\); D is the region in the first octant between planes \(z = 4 - x - y\) and \(z = 2 - x - y\).

    [Hide Solution]

    20

    Let \(F(x,y,z) = 2xi - 3xyj + xz^2k\). Use the divergence theorem to calculate \(\iint_S F \cdot dS\), where S is the surface of the cube with corners at \((0,0,0), \, (1,0,0), \, (0,1,0), \, (1,1,0), \, (0,0,1), \, (1,0,1), \, (0,1,1)\), and \((1,1,1)\), oriented outward.

    Use the divergence theorem to find the outward flux of field \(F(x,y,z) = (x^3 - 3y)i + (2yz + 1)j + xyzk\) through the cube bounded by planes \(x = \pm 1, \, y = \pm 1, \) and \(z = \pm 1\).

    [Hide Solution]

    \[\iint_S F \cdot dS = 8\]

    Let \(F(x,y,z) = 2xi - 3yj + 5zk\) and let S be hemisphere \(z = \sqrt{9 - x^2 - y^2}\) together with disk \(x^2 + y^2 \leq 9\) in the xy-plane. Use the divergence theorem.

    Evaluate \(\iint_S F \cdot N \, dS\), where \(F(x,y,z) = x^2 i + xyj + x^3y^3k\) and S is the surface consisting of all faces except the tetrahedron bounded by plane \(x + y + z = 1\)and the coordinate planes, with outward unit normal vector N.

    A vector field in three dimensions, with arrows becoming larger the further away from the origin they are, especially in their x components. S is the surface consisting of all faces except the tetrahedron bounded by the plane x + y + z = 1. As such, a portion of the given plane, the (x, y) plane, the (x, z) plane, and the (y, z) plane are shown. The arrows point towards the origin for negative x components, away from the origin for positive x components, down for positive x and negative y components, as well as positive y and negative x components, and for positive x and y components, as well as negative x and negative y components.

    [Hide Solution]

    \[\iint_S F \cdot N \, dS = \dfrac{1}{8}\]

    Find the net outward flux of field \(F = \langle bz - cy, \, cx - az, \, ay - bx \rangle\) across any smooth closed surface in \(R^3\) where a, b, and c are constants.

    Use the divergence theorem to evaluate \[\iint_S ||R||R \cdot n \, ds,\] where \(R(x,y,z) = xi + yj + zk\) and S is sphere \(x^2 + y^2 + z^2 = a^2\), with constant \(a > 0\).

    [Hide Solution]

    \[\iint_S ||R||R \cdot n \, ds = 4\pi a^4\]

    Use the divergence theorem to evaluate \[\iint_S F \cdot dS,\] where \(F(x,y,z) = y^2 zi + y^3j + xzk\) and S is the boundary of the cube defined by \(-1 \leq x \leq 1, \, -1 \leq y \leq 1\), and \(0 \leq z \leq 2\).

    Let R be the region defined by \(x^2 + y^2 + z^2 \leq 1\). Use the divergence theorem to find \[\iiint_R z^2 dV.\]

    [Hide Solution]

    \[\iiint_R z^2 dV = \dfrac{4\pi}{15}\]

    Let E be the solid bounded by the xy-plane and paraboloid \(z = 4 - x^2 - y^2\) so that S is the surface of the paraboloid piece together with the disk in the xy-plane that forms its bottom. If \(F(x,y,z) = (xz \, \sin(yz) + x^3) i + \cos (yz) j + (3zy^2 - e^{x^2+y^2})k\), find \[\iint_S F \cdot dS\] using the divergence theorem.

    A vector field in three dimensions with all of the arrows pointing down. They seem to follow the path of the paraboloid drawn opening down with vertex at the origin. S is the surface of this paraboloid and the disk in the (x, y) plane that forms its bottom.

    Let E be the solid unit cube with diagonally opposite corners at the origin and (1, 1, 1), and faces parallel to the coordinate planes. Let S be the surface of E, oriented with the outward-pointing normal. Use a CAS to find \[\iint_S F \cdot dS\] using the divergence theorem if \(F(x,y,z) = 2xyi + 3ye^zj + x \, \sin zk\).

    [Hide Solution]

    \[\iint_S F \cdot dS = 6.5759\]

    Use the divergence theorem to calculate the flux of \(F(x,y,z) = x^3i + y^3j + z^3k\) through sphere \(x^2 + y^2 + z^2 = 1\).

    Find \[\iint_S F \cdot dS,\] where \(F(x,y,z) = xi + yj + zk\) and S is the outwardly oriented surface obtained by removing cube \([1,2] \times [1,2] \times [1,2]\) from cube \([0,2] \times [0,2] \times [0,2]\).

    [Hide Solution]

    \[\iint_S F \cdot dS = 21\]

    Consider radial vector field \(F = \dfrac{r}{|r|} = \dfrac{\langle x,y,z \rangle}{(x^2+y^2+z^2)^{1/2}}\). Compute the surface integral, where S is the surface of a sphere of radius a centered at the origin.

     

    Compute the flux of water through parabolic cylinder \(S : y = x^2\), from \(0 \leq x \leq 2, \, 0 \leq z \leq 3\), if the velocity vector is \(F(x,y,z) = 3z^2i + 6j + 6xzk\).

    [Hide Solution]

    \[\iint_S F \cdot dS = 72\]

    [T} Use a CAS to find the flux of vector field \(F(x,y,z) = zi + zj + \sqrt{x^2 + y^2}k\) across the portion of hyperboloid \(x^2 + y^2 = z^2 + 1\) between planes \(z = 0\) and \(z = \dfrac{\sqrt{3}}{3}\), oriented so the unit normal vector points away from the z-axis.

    Use a CAS to find the flux of vector field \(F(x,y,z) = (e^y + x)i + (3 \, \cos (xz) - y)j + zk\) through surface S, where S is given by \(z^2 = 4x^2 + 4y^2\) from \(0 \leq z \leq 4\), oriented so the unit normal vector points downward.

    [Hide Solution]

    \[\iint_S F \cdot dS = -33.5103\]

    [T] Use a CAS to compute \[\iint_S F \cdot dS,\] where \(F(x,y,z) = xi + yj + 2zk\) and S is a part of sphere \(x^2 + y^2 + z^2 = 2\) with \(0 \leq z \leq 1\).

    Evaluate \[\iint_S F \cdot dS,\] where \(F(x,y,z) = bxy^2i + bx^2yj + (x^2 + y^2)z^2 k\) and S is a closed surface bounding the region and consisting of solid cylinder \(x^2 + y^2 \leq a^2\) and \(0 \leq z \leq b\).

    \[\iint_S F \cdot dS = \pi a^4 b^2\]

    [T] Use a CAS to calculate the flux of \(F(x,y,z) = (x^3 + y \, \sin z)i + (y^3 + z \, \sin x)j + 3zk\) across surface S, where S is the boundary of the solid bounded by hemispheres \(z = \sqrt{4 - x^2 - y^2}\) and \(z = \sqrt{1 - x^2 - y^2}\), and plane \(z = 0\).

    Use the divergence theorem to evaluate \[\iint_S F \cdot dS,\] where \(F(x,y,z) = xyi - \dfrac{1}{2}y^2j + zk\) and S is the surface consisting of three pieces: \(z = 4 - 3x^2 - 3y^2, \, 1 \leq z \leq 4\) on the top; \(x^2 + y^2 = 1, \, 0 \leq z \leq 1\) on the sides; and \(z = 0\) on the bottom.

    [Hide Solution]

    \[\iint_S F \cdot dS = \dfrac{5}{2}\pi\]

    [T] Use a CAS and the divergence theorem to evaluate \[\iint_S F \cdot dS,\] where \(F(x,y,z) = (2x + y \, \cos z)i + (x^2 - y)j + y^2 zk\) and S is sphere \(x^2 + y^2 + z^2 = 4\) orientated outward.

    Use the divergence theorem to evaluate \[\iint_S F \cdot dS,\] where \(F(x,y,z) = xi + yj + zk\) and S is the boundary of the solid enclosed by paraboloid \(y = x^2 + z^2 - 2\), cylinder \(x^2 + z^2 = 1\), and plane \(x + y = 2\), and S is oriented outward.

    [Hide Solution]

    \[\iint_S F \cdot dS = \dfrac{21\pi}{2}\]

    For the following exercises, Fourier’s law of heat transfer states that the heat flow vector F at a point is proportional to the negative gradient of the temperature; that is, \(F = - k \nabla T\), which means that heat energy flows hot regions to cold regions. The constant \(k > 0\) is called the conductivity, which has metric units of joules per meter per second-kelvin or watts per meter-kelvin. A temperature function for region D is given. Use the divergence theorem to find net outward heat flux \[\iint_S F \cdot N \, dS = -k \iint_S \nabla T \cdot N \, dS\] across the boundary S of D, where \(k = 1\).

    \(T(x,y,z) = 100 + x + 2y + z\);

    \(D = \{(x,y,z) : 0 \leq x \leq 1, \, 0 \leq y \leq 1, \, 0 \leq z \leq 1 \}\)

    \(T(x,y,z) = 100 + e^{-z}\);

    \(D = \{(x,y,z) : 0 \leq x \leq 1, \, 0 \leq y \leq 1, \, 0 \leq z \leq 1 \}\)

    [Hide Solution]

    \(- (1 - e^{-1})\)

    \(T(x,y,z) = 100 e^{-x^2-y^2-z^2}\); D is the sphere of radius a centered at the origin.

    Chapter Review Exercises

    True or False? Justify your answer with a proof or a counterexample.

    Vector field \(F(x,y) = x^2 yi + y^2 xj\) is conservative.

    [Hide Solution]

    False

    For vector field \(F(x,y) = P(x,y)i + Q(x,y)j\), if \(P_y(x,y) = Q_z(x,y)\) in open region \(D\), then \[\int_{\partial D} P dx + Qdy = 0.\]

    The divergence of a vector field is a vector field.

    [Hide Solution]

    False

    If \(curl \, F = 0\), then \(F\) is a conservative vector field.

     

    Draw the following vector fields.

    \(F(x,y) = \dfrac{1}{2}i + 2xj\)

    [Hide Solution]

    A vector field in two dimensions. All quadrants are shown. The arrows are larger the further from the y axis they become. They point up and to the right for positive x values and down and to the right for negative x values. The further from the y axis they are, the steeper the slope they have.

    \(F(x,y) = \sqrt{\dfrac{yi+3xj}{x^2+y^2}}\)

    Are the following the vector fields conservative? If so, find the potential function \(f\) such that \(F = \nabla f\).

    \(F(x,y) = yi + (x - 2e^y)j\)

    [Hide solution]

    Conservative, \(f(x,y) = xy - 2e^y\)

    \(F(x,y) = (6xy)i + (3x^2 - ye^y)j\)

    \(F(x,y) = (2xy + z^2)i + (x^2 + 2yz)j + (2xz + y^2)k\)

    [Hide Solution]

    Conservative, \(f(x,y,z) = x^2y + y^2z + z^2x\)

    \(F(x,y,z) = (e^xy)i + (e^x + z)j + (e^x + y^2)k\)

    Evaluate the following integrals.

    \[\int_C x^2 dy + (2x - 3xy) \, dx\), along \(C : y = \dfrac{1}{2}x\) from (0, 0) to (4, 2)

    [Hide solution]

    \(-\dfrac{16}{3}\)

    \(\int_C ydx + xy^2 dy\), where \(C : x = \sqrt{t}, \, y = t - 1, \, 0 \leq t \leq 1\)

    \(\iint_S xy^2 dS,\) where S is surface \(z = x^2 - y, \, 0 \leq x \leq 1, \, 0 \leq y \leq 4\)

    [Hide Solution]

    \(\dfrac{32\sqrt{2}}{9}(3\sqrt{3} - 1)\)

    Find the divergence and curl for the following vector fields.

    \(F(x,y,z) = 3xyz i + xye^x j - 3xy k\)

    \(F(x,y,z) = e^x i + e^{xy} j - e^{xyz} k\)

    [Hide Solution]

    Divergence: \(e^x + x \, e^{xy} + xy\, e^{xyz}\), curl: \(xz e^{xyz} i - yz e^{xyz}j + ye^{xy} k\)

    Use Green’s theorem to evaluate the following integrals.

    \(\int_C 3xydx + 2xy^2 dy\), where C is a square with vertices (0, 0), (0, 2), (2, 2) and (2, 0)

    \(\oint_C 3ydx + (x + e^y)dy\), where C is a circle centered at the origin with radius 3

    [Hide Solution]

    \(-2\pi\)

    Use Stokes’ theorem to evaluate \(\iint_S curl \, F \cdot dS\).

    \(F(x,y,z) = yi - xj + zk\), where \(S\) is the upper half of the unit sphere

    \(F(x,y,z) = yi + xyz j - 2zxk\), where \(S\) is the upward-facing paraboloid \(z = x^2 + y^2\) lying in cylinder \(x^2 + y^2 = 1\)

    [Hide Solution]

    \(-\pi\)

    Use the divergence theorem to evaluate \(\iint_S F \cdot dS\).

    \(F(x,y,z) = (x^3y)i + (3y - e^x)j + (z + x)k\), over cube \(S\) defined by \(-1 \leq x \leq 1, \, 0 \leq y \leq 2, \, 0 \leq z \leq 2\)

    \(F(x,y,z) = (2xy)i + (-y^2)j + (2z^3)k\), where \(S\) is bounded by paraboloid \(z = x^2 + y^2\) and plane \(z = 2\)

    [Hide Solution]

    \(31\pi /2\)

    Find the amount of work performed by a 50-kg woman ascending a helical staircase with radius 2 m and height 100 m. The woman completes five revolutions during the climb.

    Find the total mass of a thin wire in the shape of a semicircle with radius \(\sqrt{2}\), and a density function of \(\rho (x,y) = y + x^2\).

    [Hide Solution]

    \(\sqrt{2}(2 + \pi)\)

    Find the total mass of a thin sheet in the shape of a hemisphere with radius 2 for \(z \geq 0\) with a density function \(\rho (x,y,z) = x + y + z\).

    Use the divergence theorem to compute the value of the flux integral over the unit sphere with \(F(x,y,z) = 3zi + 2yj + 2xk\).

    [Hide Solution]

    \(2\pi /3\)