9.7E: EXERCISES

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 \(t\).

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.

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.

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.

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 \[\iint_S (\nabla \times F) \cdot n \, dS.\]

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