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Stokes' theorem practice problems

Stokes' theorem: a circulation around a closed curve equals the flux of the curl through a surface it bounds. 15 problems with worked solutions; in 15 of them every equation is proved by a computer algebra system.

4 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 12.370

hard
Use Stokes' theorem to evaluate \( \displaystyle \oint_C \mathbf F \cdot d\mathbf r \) for \( \displaystyle \mathbf F = \left\langle - y, x, z \right\rangle \), where \( \displaystyle C \) is the circle \( \displaystyle x^2 + y^2 = 4 \) in the plane \( \displaystyle z = 0 \), counterclockwise seen from above.
  1. Take S to be the flat disk the circle bounds, with upward normal k. Stokes: ∮_C F · dr = ∬_S (curl F) · k dA.
    Reviewed
  2. \[ \left[\begin{matrix}- \frac{d}{d z} x + \frac{d}{d y} z\\\frac{d}{d z} \left(- y\right) - \frac{d}{d x} z\\\frac{d}{d x} x - \frac{d}{d y} \left(- y\right)\end{matrix}\right] = \left[\begin{matrix}0\\0\\2\end{matrix}\right] \]
    curl F.✓ Proved
  3. \[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{2} r \left. 2 \right|_{\substack{ x=r \cos{\left(\theta \right)}\\ y=r \sin{\left(\theta \right)} }}\, dr\, d\theta = 8 \pi \]
    The k-component of the curl over the disk, in polar coordinates.✓ Proved
  4. \[ \int\limits_{0}^{2 \pi} 4\, dt = 8 \pi \]
    The circulation computed directly around C agrees.✓ Proved
Answer \( 8 \pi \)

✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 12.372

hard
Use Stokes' theorem to evaluate \( \displaystyle \oint_C \mathbf F \cdot d\mathbf r \) for \( \displaystyle \mathbf F = \left\langle y^{2}, x, z^{2} \right\rangle \), where \( \displaystyle C \) is the circle \( \displaystyle x^2 + y^2 = 9 \) in the plane \( \displaystyle z = 0 \), counterclockwise seen from above.
  1. Take S to be the flat disk the circle bounds, with upward normal k. Stokes: ∮_C F · dr = ∬_S (curl F) · k dA.
    Reviewed
  2. \[ \left[\begin{matrix}- \frac{d}{d z} x + \frac{d}{d y} z^{2}\\\frac{d}{d z} y^{2} - \frac{d}{d x} z^{2}\\\frac{d}{d x} x - \frac{d}{d y} y^{2}\end{matrix}\right] = \left[\begin{matrix}0\\0\\1 - 2 y\end{matrix}\right] \]
    curl F.✓ Proved
  3. \[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{3} r \left. 1 - 2 y \right|_{\substack{ x=r \cos{\left(\theta \right)}\\ y=r \sin{\left(\theta \right)} }}\, dr\, d\theta = 9 \pi \]
    The k-component of the curl over the disk, in polar coordinates.✓ Proved
  4. \[ \int\limits_{0}^{2 \pi} \left(- 27 \sin^{3}{\left(t \right)} + 9 \cos^{2}{\left(t \right)}\right)\, dt = 9 \pi \]
    The circulation computed directly around C agrees.✓ Proved
Answer \( 9 \pi \)

✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 12.375

hard
Use Stokes' theorem to evaluate \( \displaystyle \oint_C \mathbf F \cdot d\mathbf r \) for \( \displaystyle \mathbf F = \left\langle x z, x - y, x y \right\rangle \), where \( \displaystyle C \) is the circle \( \displaystyle x^2 + y^2 = 1 \) in the plane \( \displaystyle z = 0 \), counterclockwise seen from above.
  1. Take S to be the flat disk the circle bounds, with upward normal k. Stokes: ∮_C F · dr = ∬_S (curl F) · k dA.
    Reviewed
  2. \[ \left[\begin{matrix}\frac{\partial}{\partial y} x y - \frac{\partial}{\partial z} \left(x - y\right)\\- \frac{\partial}{\partial x} x y + \frac{\partial}{\partial z} x z\\- \frac{\partial}{\partial y} x z + \frac{\partial}{\partial x} \left(x - y\right)\end{matrix}\right] = \left[\begin{matrix}x\\x - y\\1\end{matrix}\right] \]
    curl F.✓ Proved
  3. \[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{1} r \left. 1 \right|_{\substack{ x=r \cos{\left(\theta \right)}\\ y=r \sin{\left(\theta \right)} }}\, dr\, d\theta = \pi \]
    The k-component of the curl over the disk, in polar coordinates.✓ Proved
  4. \[ \int\limits_{0}^{2 \pi} \left(\frac{\sqrt{2} \cos{\left(2 t + \frac{\pi}{4} \right)}}{2} + \frac{1}{2}\right)\, dt = \pi \]
    The circulation computed directly around C agrees.✓ Proved
Answer \( \pi \)

✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 12.378

hard
Use Stokes' theorem to evaluate \( \displaystyle \oint_C \mathbf F \cdot d\mathbf r \) for \( \displaystyle \mathbf F = \left\langle - y^{3}, x^{3}, z \right\rangle \), where \( \displaystyle C \) is the circle \( \displaystyle x^2 + y^2 = 4 \) in the plane \( \displaystyle z = 0 \), counterclockwise seen from above.
  1. Take S to be the flat disk the circle bounds, with upward normal k. Stokes: ∮_C F · dr = ∬_S (curl F) · k dA.
    Reviewed
  2. \[ \left[\begin{matrix}- \frac{d}{d z} x^{3} + \frac{d}{d y} z\\\frac{d}{d z} \left(- y^{3}\right) - \frac{d}{d x} z\\\frac{d}{d x} x^{3} - \frac{d}{d y} \left(- y^{3}\right)\end{matrix}\right] = \left[\begin{matrix}0\\0\\3 x^{2} + 3 y^{2}\end{matrix}\right] \]
    curl F.✓ Proved
  3. \[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{2} r \left. 3 x^{2} + 3 y^{2} \right|_{\substack{ x=r \cos{\left(\theta \right)}\\ y=r \sin{\left(\theta \right)} }}\, dr\, d\theta = 24 \pi \]
    The k-component of the curl over the disk, in polar coordinates.✓ Proved
  4. \[ \int\limits_{0}^{2 \pi} \left(16 \sin^{4}{\left(t \right)} + 16 \cos^{4}{\left(t \right)}\right)\, dt = 24 \pi \]
    The circulation computed directly around C agrees.✓ Proved
Answer \( 24 \pi \)

✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More stokes' theorem practice

Use Stokes' theorem to evaluate \( \displaystyle \oint_C \mathbf F \cdot d\mathbf r \) for \( \displaystyle \mathbf F = \left\langle z, x, y \right\rangle \), where \( \displaystyle C \) is the circle \( \displaystyle x^2 + y^2 = 4 \) in the plane \( \displaystyle z = 0 \), counterclockwise seen from above.
Problem 12.371hard✓ Every equation proved
Use Stokes' theorem to evaluate \( \displaystyle \oint_C \mathbf F \cdot d\mathbf r \) for \( \displaystyle \mathbf F = \left\langle z, x, y \right\rangle \), where \( \displaystyle C \) is the circle \( \displaystyle x^2 + y^2 = 1 \) in the plane \( \displaystyle z = 0 \), counterclockwise seen from above.
Problem 12.373hard✓ Every equation proved
Use Stokes' theorem to evaluate \( \displaystyle \oint_C \mathbf F \cdot d\mathbf r \) for \( \displaystyle \mathbf F = \left\langle - y, x, z \right\rangle \), where \( \displaystyle C \) is the circle \( \displaystyle x^2 + y^2 = 1 \) in the plane \( \displaystyle z = 0 \), counterclockwise seen from above.
Problem 12.374hard✓ Nihil obstat
Use Stokes' theorem to evaluate \( \displaystyle \oint_C \mathbf F \cdot d\mathbf r \) for \( \displaystyle \mathbf F = \left\langle y^{2}, x, z^{2} \right\rangle \), where \( \displaystyle C \) is the circle \( \displaystyle x^2 + y^2 = 1 \) in the plane \( \displaystyle z = 0 \), counterclockwise seen from above.
Problem 12.376hard✓ Nihil obstat
Use Stokes' theorem to evaluate \( \displaystyle \oint_C \mathbf F \cdot d\mathbf r \) for \( \displaystyle \mathbf F = \left\langle y^{2}, x, z^{2} \right\rangle \), where \( \displaystyle C \) is the circle \( \displaystyle x^2 + y^2 = 4 \) in the plane \( \displaystyle z = 0 \), counterclockwise seen from above.
Problem 12.377hard✓ Nihil obstat
Use Stokes' theorem to evaluate \( \displaystyle \oint_C \mathbf F \cdot d\mathbf r \) for \( \displaystyle \mathbf F = \left\langle x z, x - y, x y \right\rangle \), where \( \displaystyle C \) is the circle \( \displaystyle x^2 + y^2 = 9 \) in the plane \( \displaystyle z = 0 \), counterclockwise seen from above.
Problem 12.379hard✓ Nihil obstat
Use Stokes' theorem to evaluate \( \displaystyle \oint_C \mathbf F \cdot d\mathbf r \) for \( \displaystyle \mathbf F = \left\langle - y^{3}, x^{3}, z \right\rangle \), where \( \displaystyle C \) is the circle \( \displaystyle x^2 + y^2 = 9 \) in the plane \( \displaystyle z = 0 \), counterclockwise seen from above.
Problem 12.380hard✓ Nihil obstat
Use Stokes' theorem to evaluate \( \displaystyle \oint_C \mathbf F \cdot d\mathbf r \) for \( \displaystyle \mathbf F = \left\langle z, x, y \right\rangle \), where \( \displaystyle C \) is the circle \( \displaystyle x^2 + y^2 = 9 \) in the plane \( \displaystyle z = 0 \), counterclockwise seen from above.
Problem 12.381hard✓ Every equation proved
Use Stokes' theorem to evaluate \( \displaystyle \oint_C \mathbf F \cdot d\mathbf r \) for \( \displaystyle \mathbf F = \left\langle - y, x, z \right\rangle \), where \( \displaystyle C \) is the circle \( \displaystyle x^2 + y^2 = 9 \) in the plane \( \displaystyle z = 0 \), counterclockwise seen from above.
Problem 12.382hard✓ Nihil obstat
Use Stokes' theorem to evaluate \( \displaystyle \oint_C \mathbf F \cdot d\mathbf r \) for \( \displaystyle \mathbf F = \left\langle x z, x - y, x y \right\rangle \), where \( \displaystyle C \) is the circle \( \displaystyle x^2 + y^2 = 4 \) in the plane \( \displaystyle z = 0 \), counterclockwise seen from above.
Problem 12.383hard✓ Every equation proved
Use Stokes' theorem to evaluate \( \displaystyle \oint_C \mathbf F \cdot d\mathbf r \) for \( \displaystyle \mathbf F = \left\langle - y^{3}, x^{3}, z \right\rangle \), where \( \displaystyle C \) is the circle \( \displaystyle x^2 + y^2 = 1 \) in the plane \( \displaystyle z = 0 \), counterclockwise seen from above.
Problem 12.384hard✓ Nihil obstat