9.7E: EXERCISES

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Exercise $\PageIndex{1}$

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

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

2. $\vecs{F}(x,y,z) = z \, \hat{ \mathbf i} + x \, \hat{ \mathbf j} + y \, \hat{ \mathbf k}$; $S$ is hemisphere $z = (a^2 - x^2 - y^2)^{1/2}$.

Answer: \[\iint_S (curl \, \vecs{F} \cdot \vecs{N}) \, dS = \pi a^2\]

3. $\vecs{F}(x,y,z) = y^2 \, \hat{ \mathbf i}+ 2x \, \hat{ \mathbf j} + 5 \, \hat{ \mathbf k}$; $S$ is hemisphere $z = (4 - x^2 - y^2)^{1/2}$.

4. $\vecs{F}(x,y,z) = z \, \hat{ \mathbf i}+ 2x \, \hat{ \mathbf j} + 3y \, \hat{ \mathbf k}$; $S$ is upper hemisphere $z = \sqrt{9 - x^2 - y^2}$.

Answer: \[\iint_S (curl \, (\vecs{F}) \cdot \vecs{N}) \, dS = 18 \pi\]

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

6. $\vecs{F}(x,y,z) = 2y \, \hat{ \mathbf i} + 6z \, \hat{ \mathbf i} + 3x \, \hat{ \mathbf k}$; $S$ is a portion of paraboloid $z = 4 - x^2 - y^2$ and is above the $xy-$plane.

Answer: \[\iint_S (curl \, (\vecs{F}) \cdot \vecs{ N}) \, dS = -8 \pi\]

Exercise $\PageIndex{2}$

For the following exercises, use Stokes’ theorem to evaluate \[\iint_S(curl \, (\vecs{F}) \cdot \vecs{ N}) \, dS\] for the vector fields and surface.

1. $\vecs{F}(x,y,z) = xy\, \hat{ \mathbf i} - z\, \hat{ \mathbf j}$ 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.

2. $\vecs{F}(x,y,z) = xy\, \hat{ \mathbf i}+ x^2 \, \hat{ \mathbf j} + z^2\, \hat{ \mathbf k}$; and $C$ is the intersection of paraboloid $z = x^2 + y^2$ and plane $z = y$, and using the outward normal vector.

Answer: \[\iint_S (curl \, (\vecs{F}) \cdot \vecs{ N}) \, dS = 0\]

3. $\vecs{F}(x,y,z) = 4y\, \hat{ \mathbf i} + z\, \hat{ \mathbf j} + 2y \, \hat{ \mathbf k}$; and $C$ is the intersection of sphere $x^2 + y^2 + z^2 = 4$ with plane $z = 0$, and using the outward normal vector.

Exercise $\PageIndex{3}$

1. 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.

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

2. [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.

3. [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.

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

4. [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.

5. 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.

6. 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.

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

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

8. Use Stokes’ theorem for vector field $\vecs{F}(x,y,z) = z\, \hat{ \mathbf i} + 3x\, \hat{ \mathbf j} + 2z\, \hat{ \mathbf k}$ 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.

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

9. Use Stokes’ theorem for vector field $\vecs{F}(x,y,z) = - \dfrac{3}{2} y^2 \, \hat{ \mathbf i}- 2 xy\, \hat{ \mathbf j}+ yz\, \hat{ \mathbf k}$, 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.

10. 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 $R(x,y,z) = x\, \hat{ \mathbf i} + y\, \hat{ \mathbf j} + z\, \hat{ \mathbf k}$. Use Stokes’ theorem to evaluate \[\int_C(ck \times R) \cdot dS.\]

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

11. 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

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

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

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

13. Let $\vecs{F}(x,y,z) = xy\, \hat{ \mathbf i} + (e^{z^2} + y)\, \hat{ \mathbf j}+ (x + y)\, \hat{ \mathbf k}$ 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 \[\iint_S curl \, F \cdot dS.\]

14. Use Stokes’ theorem to evaluate \[ \oint F \cdot dS,\] where $\vecs{F}(x,y,z) = y\, \hat{ \mathbf i} + z\, \hat{ \mathbf j} + x\, \hat{ \mathbf k}$ and $C$ is a triangle with vertices $(0, 0, 0), (2, 0, 0) $ and $0,-2,2)$ oriented counterclockwise when viewed from above.

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

15. Use the surface integral in Stokes’ theorem to calculate the circulation of field F, $\vecs{F}(x,y,z) = x^2y^3\, \hat{ \mathbf i} + \, \hat{ \mathbf j} + z\, \hat{ \mathbf k}$ 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.

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

16. Use Stokes’ theorem to compute \[\iint_S curl \, F \cdot dS.\] where $\vecs{F}(x,y,z) = \, \hat{ \mathbf i} + xy^2\, \hat{ \mathbf j} + xy^2 \, \hat{ \mathbf k}$ 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.

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

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

18. Let $\vecs{F}(x,y,z) = xy\, \hat{ \mathbf i} + 2z\, \hat{ \mathbf j} - 2y\, \hat{ \mathbf k}$ 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.

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

19. [T] Use a CAS and let $\vecs{F}(x,y,z) = xy^2\, \hat{ \mathbf i} + (yz - x)\, \hat{ \mathbf j} + e^{yxz}\, \hat{ \mathbf k}$. 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.

20. 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.

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

21. Let $S$ be the part of paraboloid $z = 9 - x^2 - y^2$ with $z \geq 0$. Verify Stokes’ theorem for vector field $\vecs{F}(x,y,z) = 3z\, \hat{ \mathbf i} + 4x\, \hat{ \mathbf j} + 2y\, \hat{ \mathbf k}$.

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

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

23. [T] Use a CAS and Stokes’ theorem to evaluate \[\iint_S (curl \, F \cdot N) \, dS,\] where $\vecs{F}(x,y,z) = x^2 y\, \hat{ \mathbf i} + xy^2 \, \hat{ \mathbf j} + z^3 \, \hat{ \mathbf k}$ 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.

24. [T] Use a CAS and Stokes’ theorem to evaluate \[\iint_S curl \, F \cdot dS,\] where $\vecs{F}(x,y,z) = \left( \sin(y + z) - yx^2 - \dfrac{y^3}{3}\right)\, \hat{ \mathbf i} + x \, \cos (y + z) \, \hat{ \mathbf j} + \cos (2y) \, \, \hat{ \mathbf k}$ 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.

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

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

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

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

Exercise $\PageIndex{4}$

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

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

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

3. [T] Use a CAS and Stokes’ theorem to evaluate $\vecs{F}(x,y,z) = 2y\, \hat{ \mathbf i} + e^z\, \hat{ \mathbf j} - arctan \, x\, \hat{ \mathbf k}$ with $S$ as a portion of paraboloid $z = 4 - x^2 - y^2$ cut off by the $xy-$plane oriented counterclockwise.

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

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

Exercise $\PageIndex{5}$

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

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

Answer: \[A = \iint_S (\nabla \times F) \cdot n \, dS = 0\] Evaluate $ A$ using a line integral.

2. 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 $\vecs{F} = \langle 2z + y, \, 2x + z, \, 2y + x \rangle$. 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.

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

Exercise $\PageIndex{7}$

1. 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.

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

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

Answer: $C = \pi (\cos \varphi - \sin \varphi)$

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

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

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

3. 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.

4. Evaluate integral \[ \iint_S (\nabla \times F) \cdot n \, dS,\] where $\vecs{F} = - xz\, \hat{ \mathbf i} + yz \, \hat{ \mathbf j}+ xye^z \, \hat{ \mathbf k}$ 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.

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

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

a) $\vecs{F} = \nabla (x \, \sin ye^z)$

b) $\vecs{F} = \langle y^2z^3, \, z2xyz^3, 3xy^2z^2 \rangle $

Answer: 0

Exercise \(\PageIndex{1}\)

Exercise \(\PageIndex{2}\)

Exercise \(\PageIndex{3}\)

Exercise \(\PageIndex{4}\)

Exercise \(\PageIndex{5}\)