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Mathematics LibreTexts

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.

A vector field in three dimensional space. The arrows are larger the further they are from the x, y plane. The arrows curve up from below the x, y plane and slightly above it. The rest tend to curve down and horizontally. An oval-shaped curve is drawn through the middle of the space.

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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.

A diagram of an intersecting plane and ellipsoid in three dimensional space. There is an orange curve drawn to show the intersection.

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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.

A diagram in three dimensions of a vector field and the intersection of a sylinder and hemisphere. The arrows are horizontal and have negative x components for negative y components and have positive x components for positive y components. The curve of intersection between the hemisphere and cylinder is drawn in blue.

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.

A diagram of a vector field in three dimensional space showing the intersection of a plane and a cylinder. The curve where the plane and cylinder intersect is drawn in blue.

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

A diagram of a vector field in three dimensional space where a paraboloid with vertex at the origin, plane at y=0, and plane at z=4 intersect. The remaining surface is the half of a paraboloid under z=4 and above y=0.

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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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