# 15.8E: Exercises for Section 15.8


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.

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

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 $$0 \leq z \leq 9$$.

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

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 $$z = 1$$.

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

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 $$z = 1$$.

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

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.

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

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 $$(\pm 1, \, \pm 2, \, \pm 3)$$.

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

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 $$2.$$

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

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 $$0 \leq z \leq 1$$.

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?

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

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 $$z = 0$$.

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

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 $$\{(x,y,z) : x^2 + y^2 + z^2 = 6 \}$$.

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

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 $$\{(x,y,z) \,:\, x^2 + y^2 + z^2 = 4\}.$$

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

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.

$$-703.7168$$

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 $$z = 2 - x - y$$.

$$20$$

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 $$z = \pm 1$$.

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

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 $$\vecs N.$$

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

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 $$a > 0$$.

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

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 $$\displaystyle \iiint_R z^2 \, dV.$$

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

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.

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

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

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 $$[0,2] \times [0,2] \times [0,2]$$.

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

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

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

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.

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

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 $$0 \leq z \leq b$$.

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

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.

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

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.

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

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

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

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

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