# 15.7E: Exercises for Section 15.7


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 $$z = (a^2 - x^2 - y^2)^{1/2}$$.

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

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 $$z = \sqrt{9 - x^2 - y^2}$$.

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

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.

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

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.

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

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 $$t.$$

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

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.

$$\displaystyle \int_C \vecs F \cdot dS = - 9.4248$$

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.

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

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.

$$\displaystyle \iint_S curl \, \vecs F \cdot dS = 2.6667$$

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.

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

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.

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

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.

$$\displaystyle \iint_S (curl \, \vecs F \cdot \vecs N)dS = -3\pi$$

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 $$\displaystyle \int_C(c \,\mathbf{\hat k} \times \vecs R) \cdot dS.$$

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

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 $$\displaystyle \iint_S curl \, \vecs F \cdot dS.$$

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

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.

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

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.

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.

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

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.

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

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.

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

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 $$x = \cos t, \, y = \sin t, \, z = 1; \, 0 \leq t \leq 2\pi$$.

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

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 $$\vecs r(r,\theta) = r \, \cos \theta \,\mathbf{\hat i} + r \, \sin \theta \,\mathbf{\hat j} + (4 - r^2) \,\mathbf{\hat k} \, (0 \leq \theta \leq 2\pi, \, 0 \leq r \leq 3)$$.

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

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 $$A.$$

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

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

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

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 $$\varphi$$?

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

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.

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

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 $$S.$$

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

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

$$0$$