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9.1E: Exercises

  • Page ID
    26272
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    Exercise \(\PageIndex{1}\)

    For the following exercises, determine whether the statement is true or false.

    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 \(F\) is a set of vectors in the plane.

    Answer

    True.

    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.

    Exercise \(\PageIndex{2}\)

    For the following exercises, 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

    A visual representation of a vector field in two dimensions. The arrows are larger the further they are from the origin. They stretch away from the origin in a radial pattern.

    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

    A visual representation of a vector field in two dimensions. The arrows are larger the further they are from the origin and the further to the left and right they are from the y axis. The arrows asymptotically curve down and to the right in quadrant 1, down and to the left in quadrant 2, up and to the left in quadrant 3, and up and to the right in quadrant four.

    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

    A visual representation of a vector field in two dimensions. The arrows are larger the further away from the origin they are and, even more so, the further away from the y axis they are. They stretch out away from the origin in a radial manner.

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

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

    Answer

    A visual representation of a vector field in two dimensions. The arrows are larger the further away they are from the x axis. The arrows form two radial patterns, one on each side of the y axis. The patterns are clockwise.

    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

    A visual representation of a vector field in three dimensions. The arrows seem to get smaller as both the z component gest close to zero and the x component gets larger, and as both the y and z components get larger. The arrows seem to converge in both of those directions as well.

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

    Exercise \(\PageIndex{3}\)

    For the following exercises, find the gradient vector field \(\vecs{F}\) 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. \(f(x,y)=x^2sin(5y)\)

    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. \(f(x,y,z)=xcos(\dfrac{y}{z})\)

    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

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

    Exercise \(\PageIndex{4}\)

    For the following exercises, 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 a constant length.

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

    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 \(\vecs{F}=0\) 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 \(|\vecs{F}|=\sqrt{a^2+b^2}\).

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

    26. Is vector field \(\vecs{F}(x,y)=(P(x,y),Q(x,y))=(sinx+y)\, \hat{\mathbf i}+(cosy+x)\, \hat{\mathbf j}\) 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

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

    Exercise \(\PageIndex{5}\)

    For the following exercises, 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 \(V(x,y)=cln(\dfrac{r_0}{\sqrt{x^2+y^2}})\), 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 \(\vecs{E}(x,y)=−∇V(x,y).\)

    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

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

    Exercise \(\PageIndex{6}\)

    A flow line (or streamline) of a vector field\vecs{F}is a curve r(t) such that \(dr/dt=\vecs{F}(r(t))\). 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 the following exercises, show that the given curve \(c(t)\) is a flow line of the given velocity vector field \(\vecs{F}(x,y,z)\).

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

    31. \(c(t)=(sint,cost,e^t);\vecs{F}(x,y,z)=〈y,−x,z〉\)

    Answer

    \(c′(t)=(cost,−sint,e^{−t})=\vecs{F}(c(t))\)

    Exercise \(\PageIndex{7}\)

    For the following exercises, 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.

    A visual representation of a vector field in two dimensions. The arrows circle the origin in a counterclockwise manner. The arrows are larger the further they are from the origin.

    33.

    A visual representation of a vector field in two dimensions. The arrows are larger the further away from the origin they are, particularly to the left and right of the y axis. They point to the left and to the right on the left and right sides of the y axis, respectively. They point down above the x axis and up below the x axis. The closer to the x axis, the flatter they are. The closer to the y axis they are, the more vertical they are.

    Answer

    \(\vecs{H}\)

    34.

    A visual representation of a vector field in two dimensions. The arrows are larger the further away from the origin they are. They stretch out and away from the origin in a radial pattern.

    Answer

    Add texts here. Do not delete this text first.

    Exercise \(\PageIndex{8}\)

    For the following exercises, 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. \( \vecs{H}+ \vecs{G}\)
    4. \( \vecs{-F}+ \vecs{G}\)

    35.

    A visual representation of a vector field in two dimensions. The arrows are larger the further away they are from the y axis. They curve in towards the orgin in a spiral pattern, with the arrows curving in from the right to the right of the y axis and curving in from the left to the left of the y axis. The arrows in quadrants 1 and 3 are flatter, and the arrows in quadrants 2 and 4 are more vertical.

    Answer

    \( \vecs{-F}+ \vecs{G}\)

    36.

    A visual representation of a vector field in two dimensions. The arrows are larger the further away from the y axis they are. They are completely flat and point to the right on the right side of the y axis and point to the left on the left side of the y axis.

    37.

    A visual representation of a vector field in two dimensions. The arrows are larger the further away from y axis they are. The arrows are pointing out from the origin in a spiral shape. In quadrant 1, the arrows are more vertical and curve up. In quadrant 2, the arrows are more horizontal and curve down. In quadrant 3, the arrows are more vertical and curve down. In quadrant 4, the arrows are more horizontal and curve up. Each quadrant’s arrows merge into the two on either side of it.

    Answer

    \( \vecs{F}+ \vecs{G}\)

    38.

    A visual representation of a vector field in two dimensions. The arrows are larger the further away they are from the x axis and y axis in quadrants 2 and 4. The arrows are all at a roughly 90-degree angle. They point up on the right side of the y axis and down on the left side of the y axis.


    This page titled 9.1E: Exercises is shared under a not declared license and was authored, remixed, and/or curated by Pamini Thangarajah.

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