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# 16.E: Vector Calculus (Exercises)

16.1: Vector Fields

## 16.7: Stokes’ Theorem

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

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

$$F(x,y,z) = zi + xj + yk$$; S is hemisphere $$z = (a^2 - x^2 - y^2)^{1/2}$$.

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

$$F(x,y,z) = y^2i + 2xj + 5k$$; S is hemisphere $$z = (4 - x^2 - y^2)^{1/2}$$.

$$F(x,y,z) = zi + 2xj + 3yk$$; S is upper hemisphere $$z = \sqrt{9 - x^2 - y^2}$$.

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

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

$$F(x,y,z) = 2yi + 6zj + 3xk$$; S is a portion of paraboloid $$z = 4 - x^2 - y^2$$ and is above the xy-plane.

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

For the following exercises, use Stokes’ theorem to evaluate

$\iint_S (curl \, F \cdot N) \, dS$ for the vector fields and surface.

$$F(x,y,z) = xyi - zj$$ 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.

$$F(x,y,z) = xyi + x^2 j + z^2 k$$; and C is the intersection of paraboloid $$z = x^2 + y^2$$ and plane $$z = y$$, and using the outward normal vector.

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

$$F(x,y,z) = 4yi + z j + 2y k$$; and C is the intersection of sphere $$x^2 + y^2 + z^2 = 4$$ with plane $$z = 0$$, and using the outward normal vector.

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.

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$\int_C F \cdot dS = 0$

[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.

[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.

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$\int_C F \cdot dS = - 9.4248$

[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.

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

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$\iint_S F \cdot dS = 0$

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

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

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

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

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

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

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.

Use Stokes’ theorem to evaluate $\int_C \left(\frac{1}{2} y^2 \, dx + z \, dy + x \, dz \right),$ 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.

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

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

Use Stokes’ theorem for vector field $$F(x,y,z) = zi + 3xj + 2zk$$ 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.

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$\iint_S (curl \, F \cdot N)dS = -3\pi] Use Stokes’ theorem for vector field $$F(x,y,z) = - \frac{3}{2} y^2 i - 2 xyj + yzk$$, 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. 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 $$R(x,y,z) = xi + yj + zk$$. Use Stokes’ theorem to evaluate \[\int_C(ck \times R) \cdot dS.$

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$\int_C (ck \times R) \cdot dS = 2\pi c$

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

$$F(x,y,z) = [\sin (x^3) + xz] i + (x - yz)j + \cos (z^4) k$$ and evaluate $$\int_C F \cdot dS$$.

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

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

Let $$F(x,y,z) = xyi + (e^{z^2} + y)j + (x + y)k$$ and let S be the graph of function $$y = \frac{x^2}{9} + \frac{z^2}{9} - 1$$ with $$z \leq 0$$ oriented so that the normal vector S has a positive y component. Use Stokes’ theorem to compute integral $\iint_S curl \, F \cdot dS.$

Use Stokes’ theorem to evaluate $\oint F \cdot dS,$ where $$F(x,y,z) = yi + zj + xk$$ and C is a triangle with vertices (0, 0, 0), (2, 0, 0) and $$0,-2,2)$$ oriented counterclockwise when viewed from above.

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$\oint F \cdot dS = -4$

Use the surface integral in Stokes’ theorem to calculate the circulation of field F, $$F(x,y,z) = x^2y^3 i + j + zk$$ 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.

Use Stokes’ theorem to compute $\iint_S curl \, F \cdot dS.$ where $$F(x,y,z) = i + xy^2j + xy^2 k$$ and S is a part of plane $$y + z = 2$$  inside cylinder $$x^2 + y^2 = 1$$ and oriented counterclockwise.

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

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

Let $$F(x,y,z) = xyi + 2zj - 2yk$$ 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.

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

[T] Use a CAS and let $$F(x,y,z) = xy^2i + (yz - x)j + e^{yxz}k$$. 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.

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

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

Let S be the part of paraboloid $$z = 9 - x^2 - y^2$$ with $$z \geq 0$$. Verify Stokes’ theorem for vector field $$F(x,y,z) = 3zi + 4xj + 2yk$$.

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

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$\oint_C F \cdot dr = 0$

[T] Use a CAS and Stokes’ theorem to evaluate $$F(x,y,z) = 2yi + e^zj - arctan \, xk$$ with S as a portion of paraboloid $$z = 4 - x^2 - y^2$$ cut off by the xy-plane oriented counterclockwise.

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

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

Let S be paraboloid $$z = a (1 - x^2 - y^2)$$, for $$z \geq 0$$, where $$a > 0$$ is a real number. Let $$F(x,y,z) = \langle x - y, \, y + z, \, z - x \rangle$$. For what value(s) of a (if any) does $\iint_S (\nabla \times F) \cdot n \, dS$ have its maximum value?

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

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

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

Evaluate A using a line integral.

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 $$F = \langle 2z + y, \, 2x + z, \, 2y + x \rangle$$. Evaluate $\iint_S (\nabla \times F) \cdot n \, dS.$

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

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 $$0 \leq \varphi \leq \frac{\pi}{2}$$ is a fixed angle.

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

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

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

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

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

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$\oint_C F \cdot dr = 48 \pi$

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.

Evaluate integral $\iint_S (\nabla \times F) \cdot n \, dS,$ where $$F = - xzi + yzj + xye^z k$$ 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.

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

For the following exercises, use Stokes’ theorem to find the circulation of the following vector fields around any smooth, simple closed curve C.

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

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

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