15.7E: Exercises for Section 15.7

Last updated
Save as PDF

Page ID: 67003

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $$\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$

In exercises 1 - 6, without using Stokes’ theorem, calculate directly both the flux of $curl \, \vecs F \cdot \vecs N$ over the given surface and the circulation integral around its boundary, assuming all are oriented clockwise.

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

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

Answer: $\displaystyle \iint_S (curl \, \vecs F \cdot \vecs N) \, dS = \pi a^2$

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

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

Answer: $\displaystyle \iint_S (curl \, \vecs F \cdot \vecs N) \, dS = 18 \pi$

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

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

Answer: $\displaystyle \iint_S (curl \, \vecs F \cdot \vecs N) \, dS = -8 \pi$

In exercises 7 - 9, use Stokes’ theorem to evaluate $\displaystyle \iint_S (curl \, \vecs F \cdot \vecs N) \, dS$ for the vector fields and surface.

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

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

Answer: $\displaystyle \iint_S (curl \, \vecs F \cdot \vecs N) \, dS = 0$

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

10. Use Stokes’ theorem to evaluate $\displaystyle \int_C \big[2xy^2z \, dx + 2x^2yz \, dy + (x^2y^2 - 2z) \, dz\big],$ 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: $\displaystyle \int_C \vecs F \cdot dS = 0$

11. [T] Use a computer algebraic system (CAS) and Stokes’ theorem to approximate line integral $\displaystyle \int_C (y \, dx + z \, dy + x \, dz),$ 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.

12. [T] Use a CAS and Stokes’ theorem to approximate line integral $\displaystyle \int_C (3y\, dx + 2z \, dy - 5x \, dz),$ 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: $\displaystyle \int_C \vecs F \cdot dS = - 9.4248$

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

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

Answer: $\displaystyle \iint_S \vecs F \cdot dS = 0$

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

16. [T] Use a CAS and Stokes’ theorem to evaluate $\displaystyle \iint_S curl \, \vecs F \cdot dS,$ where $\vecs F(x,y,z) = \left( \sin(y + z) - yx^2 - \dfrac{y^3}{3}\right)\,\mathbf{\hat i} + x \, \cos (y + z) \,\mathbf{\hat j} + \cos (2y) \,\mathbf{\hat k}$ 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: $\displaystyle \iint_S curl \, \vecs F \cdot dS = 2.6667$

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

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

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

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

20. Use Stokes’ theorem to evaluate $\displaystyle \int_C \left(\dfrac{1}{2} y^2 \, dx + z \, dy + x \, dz \right),$ 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: $\displaystyle \int_C \left(\dfrac{1}{2} y^2 \, dx + z \, dy + x \, dz \right) = - \dfrac{\pi}{4}$

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

22. Use Stokes’ theorem for vector field $\vecs F(x,y,z) = z\,\mathbf{\hat i} + 3x\,\mathbf{\hat j} + 2z\,\mathbf{\hat k}$ 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: $\displaystyle \iint_S (curl \, \vecs F \cdot \vecs N)dS = -3\pi$

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

24. 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 $\vecs R(x,y,z) = x\,\mathbf{\hat i} + y\,\mathbf{\hat j} + z\,\mathbf{\hat k}$. Use Stokes’ theorem to evaluate $\displaystyle \int_C(c \,\mathbf{\hat k} \times \vecs R) \cdot dS.$

Answer: $\displaystyle \int_C (c \,\mathbf{\hat k} \times \vecs R) \cdot dS = 2\pi c$

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

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

Answer: $\displaystyle \iint_S curl \, \vecs F \cdot dS = 0$

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

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

Answer: $\displaystyle \oint \vecs F \cdot dS = -4$

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

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

30. Use Stokes’ theorem to compute $\displaystyle \iint_S curl \, \vecs F \cdot dS.$ where $\vecs F(x,y,z) = \,\mathbf{\hat i} + xy^2\,\mathbf{\hat j} + xy^2 \,\mathbf{\hat k}$ 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: $\displaystyle \iint_S curl \, \vecs F \cdot dS = 0$

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

32. Let $\vecs F(x,y,z) = xy\,\mathbf{\hat i} + 2z\,\mathbf{\hat j} - 2y\,\mathbf{\hat k}$ 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 $\vecs F$ over $C$ using Stokes’ theorem.

Answer: $\displaystyle \iint_S curl \, \vecs F \cdot dS = -36 \pi$

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

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

Answer: $\displaystyle \iint_S curl \, \vecs F \cdot \vecs N = 0$

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

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

Answer: $\displaystyle \oint_C \vecs F \cdot d\vecs{r} = 0$

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

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

Answer: $\displaystyle \iint_S curl (F) \cdot dS = 84.8230$

39. 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 $\displaystyle \iint_S (\vecs \nabla \times \vecs F) \cdot \vecs n \, dS$ have its maximum value?

For application exercises 40 - 41, the goal is to evaluate $\displaystyle A = \iint_S (\vecs \nabla \times \vecs F) \cdot \vecs n \, dS,$ where $\vecs F = \langle xz, \, -xz, \, xy \rangle$ and $S$ is the upper half of ellipsoid $x^2 + y^2 + 8z^2 = 1$, where $z \geq 0$.

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

Answer: $\displaystyle A = \iint_S (\vecs \nabla \times \vecs F) \cdot \vecs n \, dS = 0$

41. Evaluate $A$ using a line integral.

42. 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 $\displaystyle \iint_S (\vecs \nabla \times \vecs F) \cdot \vecs n \, dS.$

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

Answer: $\displaystyle \iint_S (\vecs \nabla \times \vecs F) \cdot \vecs n \, dS = 2\pi$

For exercises 43 - 45, let $S$ be the disk enclosed by curve $C \, : \, \vecs r(t) = \langle \cos \varphi \, \cos t, \, \sin t, \, \sin \varphi \, \cos t \rangle$, for $0 \leq t \leq 2\pi$, where $0 \leq \varphi \leq \dfrac{\pi}{2}$ is a fixed angle.

43. What is the length of $C$ in terms of $\varphi$?

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

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

45. For what value of $\varphi$ is the circulation a maximum?

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

Answer: $\displaystyle \oint_C \vecs F \cdot d\vecs{r} = 48 \pi$

47. 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 $\vecs v$ in a clockwise rotation.

48. Evaluate integral $\displaystyle \iint_S (\vecs \nabla \times \vecs F) \cdot \vecs n \, dS,$ where $\vecs F = - xz\,\mathbf{\hat i} + yz\,\mathbf{\hat j} + xye^z \,\mathbf{\hat k}$ and $S$ is the cap of paraboloid $z = 5 - x^2 - y^2$ above plane $z = 3$, and $\vecs n$ points in the positive $z$-direction on $S.$

Answer: $\displaystyle \iint_S (\vecs \nabla \times \vecs F) \cdot \vecs n = 0$

In exercises 49 - 50, use Stokes’ theorem to find the circulation of the following vector fields around any smooth, simple closed curve $C.$

49. $\vecs F = \vecs \nabla (x \, \sin ye^z)$

50. $\vecs F = \langle y^2z^3, \, z2xyz^3, 3xy^2z^2 \rangle $

Answer: $0$

Contributors

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.