16.8E: Exercises for Section 16.8

Last updated
Save as PDF

Page ID: 67036

Gilbert Strang & Edwin “Jed” Herman
OpenStax

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $$\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$

For exercises 1 - 9, use a computer algebraic system (CAS) and the divergence theorem to evaluate surface integral $\displaystyle \int_S \vecs F \cdot \vecs n \, ds$ for the given choice of $\vecs F$ and the boundary surface $S.$ For each closed surface, assume $\vecs N$ is the outward unit normal vector.

1. [T] $\vecs F(x,y,z) = x\,\mathbf{\hat i} + y\,\mathbf{\hat j} + z\,\mathbf{\hat k}$; $S$ is the surface of cube $0 \leq x \leq 1, \, 0 \leq y \leq 1, \, 0 < z \leq 1$.

2. [T] $\vecs F(x,y,z) = (\cos yz) \,\mathbf{\hat i} + e^{xz}\,\mathbf{\hat j} + 3z^2 \,\mathbf{\hat k}$; $S$ 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.

Answer: $\displaystyle \int_S \vecs F \cdot \vecs n \, ds = 75.3982$

3. [T] $\vecs F(x,y,z) = (x^2 + y^2 - x^2)\,\mathbf{\hat i} + x^2 y\,\mathbf{\hat j} + 3z\,\mathbf{\hat k}; $ $S$ is the surface of the five faces of unit cube $0 \leq x \leq 1, \, 0 \leq y \leq 1, \, 0 < z \leq 1.$

4. [T] $\vecs F(x,y,z) = x\,\mathbf{\hat i} + y\,\mathbf{\hat j} + z\,\mathbf{\hat k}; $ $S$ is the surface of paraboloid $z = x^2 + y^2$ for $0 \leq z \leq 9$.

Answer: $\displaystyle \int_S \vecs F \cdot \vecs n \, ds = 127.2345$

5. [T] $\vecs F(x,y,z) = x^2\,\mathbf{\hat i} + y^2 \,\mathbf{\hat j} + z^2 \,\mathbf{\hat k}$; $S$ is the surface of sphere $x^2 + y^2 + z^2 = 4$.

6. [T] $\vecs F(x,y,z) = x\,\mathbf{\hat i} + y\,\mathbf{\hat j} + (z^2 - 1)\,\mathbf{\hat k}$; $S$ is the surface of the solid bounded by cylinder $ x^2 + y^2 = 4$ and planes $z = 0$ and $z = 1$.

Answer: $\displaystyle \int_S \vecs F \cdot \vecs n \, ds = 37.699$

7. [T] $\vecs F(x,y,z) = x^2\,\mathbf{\hat i} + y^2 \,\mathbf{\hat j} + z^2 \,\mathbf{\hat k}$; $S$ 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.”)

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

Answer: $\displaystyle \int_S \vecs F \cdot \vecs n \, ds = \dfrac{9\pi a^4}{2}$

9. [T] Surface integral $\displaystyle \iint_S \vecs F \cdot dS$, where $S$ is the solid bounded by paraboloid $z = x^2 + y^2$ and plane $z = 4$, and $\vecs F(x,y,z) = (x + y^2z^2)\,\mathbf{\hat i} + (y + z^2x^2)\,\mathbf{\hat j} + (z + x^2y^2) \,\mathbf{\hat k}$

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

Answer: $\displaystyle \iint_S \vecs F \cdot dS = \dfrac{\pi}{3}$

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

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

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

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

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

Answer: $\displaystyle \iint_S \vecs F \cdot dS = 241.2743$

15. Use the divergence theorem to compute the value of flux integral $\displaystyle \iint_S \vecs F \cdot dS$, where $\vecs F(x,y,z) = (y^3 + 3x)\,\mathbf{\hat i} + (xz + y)\,\mathbf{\hat j} + \left(z + x^4 \cos (x^2y)\right)\,\mathbf{\hat k}$ and $S$ is the area of the region bounded by $x^2 + y^2 = 1, \, x \geq 0, \, y \geq 0$, and $0 \leq z \leq 1$.

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

16. Use the divergence theorem to compute flux integral $\displaystyle \iint_S \vecs F \cdot dS$, where $\vecs F(x,y,z) = y\,\mathbf{\hat j} - z\,\mathbf{\hat k}$ 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?

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

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

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

Answer: $\displaystyle \iint_S \vecs F \cdot dS = \dfrac{2\pi}{3}$

19. Consider $\vecs F(x,y,z) = x^2\,\mathbf{\hat i} + xy\,\mathbf{\hat j} + (z + 1)\,\mathbf{\hat k}$. 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 $\vecs F$ across the boundary of $E$ using the divergence theorem.

In exercises 20 - 23, use a CAS along with the divergence theorem to compute the net outward flux for the fields across the given surfaces $S.$

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

Answer: $15\sqrt{6}\pi$

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

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

Answer: $-\dfrac{128}{3} \pi$

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

For exercises 24 - 26, use a CAS and the divergence theorem to compute the net outward flux for the vector fields across the boundary of the given regions $D.$

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

Answer: $-703.7168$

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

26. [T] $\vecs F = \langle x^2, \, -y^2, \, z^2 \rangle$; $D$ is the region in the first octant between planes $z = 4 - x - y$ and $z = 2 - x - y$.

Answer: $20$

27. Let $\vecs F(x,y,z) = 2x\,\mathbf{\hat i} - 3xy\,\mathbf{\hat j} + xz^2\,\mathbf{\hat k}$. Use the divergence theorem to calculate $\displaystyle \iint_S \vecs F \cdot dS$, 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.

28. Use the divergence theorem to find the outward flux of field $\vecs F(x,y,z) = (x^3 - 3y)\,\mathbf{\hat i} + (2yz + 1)\,\mathbf{\hat j} + xyz\,\mathbf{\hat k}$ through the cube bounded by planes $x = \pm 1, \, y = \pm 1, $ and $z = \pm 1$.

Answer: $\displaystyle \iint_S \vecs F \cdot dS = 8$

29. Let $\vecs F(x,y,z) = 2x\,\mathbf{\hat i} - 3y\,\mathbf{\hat j} + 5z\,\mathbf{\hat k}$ 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.

30. Evaluate $\displaystyle \iint_S \vecs F \cdot \vecs n \, dS$, where $\vecs F(x,y,z) = x^2 \,\mathbf{\hat i} + xy\,\mathbf{\hat j} + x^3y^3\,\mathbf{\hat k}$ 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 $\vecs N.$

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

Answer: $\displaystyle \iint_S \vecs F \cdot \vecs n \, dS = \dfrac{1}{8}$

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

32. Use the divergence theorem to evaluate $\displaystyle \iint_S ||\vecs R||\vecs R \cdot \vecs n \, ds,$ where $\vecs R(x,y,z) = x\,\mathbf{\hat i} + y\,\mathbf{\hat j} + z\,\mathbf{\hat k}$ and $S$ is sphere $x^2 + y^2 + z^2 = a^2$, with constant $a > 0$.

Answer: $\displaystyle \iint_S ||\vecs R||\vecs R \cdot \vecs n \, ds = 4\pi a^4$

33. Use the divergence theorem to evaluate $\displaystyle \iint_S \vecs F \cdot dS,$ where $\vecs F(x,y,z) = y^2 z\,\mathbf{\hat i} + y^3\,\mathbf{\hat j} + xz\,\mathbf{\hat k}$ and $S$ is the boundary of the cube defined by $-1 \leq x \leq 1, \, -1 \leq y \leq 1$, and $0 \leq z \leq 2$.

34. Let $R$ be the region defined by $x^2 + y^2 + z^2 \leq 1$. Use the divergence theorem to find $\displaystyle \iiint_R z^2 \, dV.$

Answer: $\displaystyle \iiint_R z^2 dV = \dfrac{4\pi}{15}$

35. 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 $\vecs F(x,y,z) = (xz \, \sin(yz) + x^3) \,\mathbf{\hat i} + \cos (yz) \,\mathbf{\hat j} + (3zy^2 - e^{x^2+y^2})\,\mathbf{\hat k}$, find $\displaystyle \iint_S \vecs F \cdot dS$ using the divergence theorem.

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

36. 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 $\displaystyle \iint_S \vecs F \cdot dS$ using the divergence theorem if $\vecs F(x,y,z) = 2xy\,\mathbf{\hat i} + 3ye^z\,\mathbf{\hat j} + x \sin z\,\mathbf{\hat k}$.

Answer: $\displaystyle \iint_S \vecs F \cdot dS = 6.5759$

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

38. Find $\displaystyle \iint_S \vecs F \cdot dS,$ where $\vecs F(x,y,z) = x\,\mathbf{\hat i} + y\,\mathbf{\hat j} + z\,\mathbf{\hat k}$ and $S$ is the outwardly oriented surface obtained by removing cube $[1,2] \times [1,2] \times [1,2]$ from cube $[0,2] \times [0,2] \times [0,2]$.

Answer: $\displaystyle \iint_S \vecs F \cdot dS = 21$

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

40. 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 $\vecs F(x,y,z) = 3z^2\,\mathbf{\hat i} + 6\,\mathbf{\hat j} + 6xz\,\mathbf{\hat k}$.

Answer: $\displaystyle \iint_S \vecs F \cdot dS = 72$

41. [T] Use a CAS to find the flux of vector field $\vecs F(x,y,z) = z\,\mathbf{\hat i} + z\,\mathbf{\hat j} + \sqrt{x^2 + y^2}\,\mathbf{\hat k}$ 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.

42. Use a CAS to find the flux of vector field $\vecs F(x,y,z) = (e^y + x)\,\mathbf{\hat i} + (3 \, \cos (xz) - y)\,\mathbf{\hat j} + z\,\mathbf{\hat k}$ 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.

Answer: $\displaystyle \iint_S \vecs F \cdot dS = -33.5103$

43. [T] Use a CAS to compute $\displaystyle \iint_S \vecs F \cdot dS,$ where $\vecs F(x,y,z) = x\,\mathbf{\hat i} + y\,\mathbf{\hat j} + 2z\,\mathbf{\hat k}$ and $S$ is a part of sphere $x^2 + y^2 + z^2 = 2$ with $0 \leq z \leq 1$.

44. Evaluate $\displaystyle \iint_S \vecs F \cdot dS,$ where $\vecs F(x,y,z) = bxy^2\,\mathbf{\hat i} + bx^2y\,\mathbf{\hat j} + (x^2 + y^2)z^2 \,\mathbf{\hat k}$ and $S$ is a closed surface bounding the region and consisting of solid cylinder $x^2 + y^2 \leq a^2$ and $0 \leq z \leq b$.

Answer: $\displaystyle \iint_S \vecs F \cdot dS = \pi a^4 b^2$

45. [T] Use a CAS to calculate the flux of $\vecs F(x,y,z) = (x^3 + y \, \sin z)\,\mathbf{\hat i} + (y^3 + z \, \sin x)\,\mathbf{\hat j} + 3z\,\mathbf{\hat k}$ 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 $z = 0$.

46. Use the divergence theorem to evaluate $\displaystyle \iint_S \vecs F \cdot dS,$ where $\vecs F(x,y,z) = xy\,\mathbf{\hat i} - \dfrac{1}{2}y^2\,\mathbf{\hat j} + z\,\mathbf{\hat k}$ 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.

Answer: $\displaystyle \iint_S \vecs F \cdot dS = \dfrac{5}{2}\pi$

47. [T] Use a CAS and the divergence theorem to evaluate $\displaystyle \iint_S \vecs F \cdot dS,$ where $\vecs F(x,y,z) = (2x + y \, \cos z)\,\mathbf{\hat i} + (x^2 - y)\,\mathbf{\hat j} + y^2 z\,\mathbf{\hat k}$ and $S$ is sphere $x^2 + y^2 + z^2 = 4$ orientated outward.

48. Use the divergence theorem to evaluate $\displaystyle \iint_S \vecs F \cdot dS,$ where $\vecs F(x,y,z) = x\,\mathbf{\hat i} + y\,\mathbf{\hat j} + z\,\mathbf{\hat k}$ 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.

Answer: $\displaystyle \iint_S \vecs F \cdot dS = \dfrac{21\pi}{2}$

For the following exercises, Fourier’s law of heat transfer states that the heat flow vector $\vecs F$ at a point is proportional to the negative gradient of the temperature; that is, $\vecs F = - k \vecs \nabla T$, 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 $D$ s given. Use the divergence theorem to find net outward heat flux $\displaystyle \iint_S \vecs F \cdot \vecs n \, dS = -k \iint_S \vecs \nabla T \cdot N \, dS$ across the boundary $S$ of $D,$ where $k = 1$.

49. $T(x,y,z) = 100 + x + 2y + z$;

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

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

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

Answer: $- (1 - e^{-1})$

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

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.