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Mathematics LibreTexts

16R: Chapter 16 Review Exercises

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    67080
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    True or False?  Justify your answer with a proof or a counterexample.

    1. The vector field \(\vecs F(x,y) = x^2 y\,\mathbf{\hat i} + y^2 x\,\mathbf{\hat j}\) is conservative.

    Answer
    False

    2. For vector field \(\vecs F(x,y) = P(x,y)\,\mathbf{\hat i} + Q(x,y)\,\mathbf{\hat j} \), if \(P_y(x,y) = Q_z(x,y)\) in open region \(D\), then \(\displaystyle \int_{\partial D} P \,dx + Q \, dy = 0.\)

    3. The divergence of a vector field is a vector field.

    Answer
    False

    4. If \(curl \, \vecs F = \vecs 0\), then \(\vecs F\) is a conservative vector field.

     

    Draw the following vector fields.

    5. \(\vecs F(x,y) = \dfrac{1}{2}\,\mathbf{\hat i} + 2x\,\mathbf{\hat j} \)

    Answer
    A vector field in two dimensions. All quadrants are shown. The arrows are larger the further from the y axis they become. They point up and to the right for positive x values and down and to the right for negative x values. The further from the y axis they are, the steeper the slope they have.

    6. \(\vecs F(x,y) = \sqrt{\dfrac{y\,\mathbf{\hat i}+3x\,\mathbf{\hat j}}{x^2+y^2}}\)

     

    Are the following the vector fields conservative? If so, find the potential function \(\vecs F\) such that \(\vecs F =  \vecs \nabla f\).

    7. \(\vecs F(x,y) = y\,\mathbf{\hat i} + (x - 2e^y)\,\mathbf{\hat j} \)

    Answer
    Conservative, \(f(x,y) = xy - 2e^y\)

    8. \(\vecs F(x,y) = (6xy)\,\mathbf{\hat i} + (3x^2 - ye^y)\,\mathbf{\hat j} \)

    9. \(\vecs F(x,y) = (2xy + z^2)\,\mathbf{\hat i} + (x^2 + 2yz)\,\mathbf{\hat j} + (2xz + y^2)\,\mathbf{\hat k} \)

    Answer
    Conservative, \(f(x,y,z) = x^2y + y^2z + z^2x\)

    10. \(\vecs F(x,y,z) = (e^xy)\,\mathbf{\hat i} + (e^x + z)\,\mathbf{\hat j} + (e^x + y^2)\,\mathbf{\hat k} \)

     

    Evaluate the following integrals.

    11. \(\displaystyle \int_C x^2 \, dy + (2x - 3xy) \, dx\), along \(C : y = \dfrac{1}{2}x\) from \((0, 0)\) to \((4, 2)\)

    Answer
    \(-\dfrac{16}{3}\)

    12. \(\displaystyle \int_C y\, dx + xy^2 \, dy\), where \(C : x = \sqrt{t}, \, y = t - 1, \, 0 \leq t \leq 1\)

    13. \(\displaystyle \iint_S xy^2 \, dS,\) where \(S\) is the surface \(z = x^2 - y, \, 0 \leq x \leq 1, \, 0 \leq y \leq 4\)

    Answer
    \(\dfrac{32\sqrt{2}}{9}(3\sqrt{3} - 1)\)

     

    Find the divergence and curl for the following vector fields.

    14. \(\vecs F(x,y,z) = 3xyz \,\mathbf{\hat i} + xye^x \,\mathbf{\hat j} - 3xy \,\mathbf{\hat k} \)

    15. \(\vecs F(x,y,z) = e^x \,\mathbf{\hat i}  + e^{xy} \,\mathbf{\hat j} - e^{xyz} \,\mathbf{\hat k} \)

    Answer
    Divergence: \(e^x + x \, e^{xy} + xy\, e^{xyz}\)
    Curl: \(xz e^{xyz} \,\mathbf{\hat i} - yz e^{xyz} \,\mathbf{\hat j}  + ye^{xy} \,\mathbf{\hat k} \)

     

    Use Green’s theorem to evaluate the following integrals.

    16. \(\displaystyle \int_C 3xy \, dx + 2xy^2 \, dy\), where \(C\) is a square with vertices \((0, 0), \, (0, 2), \, (2, 2)\) and \((2, 0).\)

    17. \(\displaystyle \oint_C 3y\, dx + (x + e^y)\, dy\), where \(C\) is a circle centered at the origin with radius \(3.\)

    Answer
    \(-2\pi\)

     

    Use Stokes’ theorem to evaluate \(\iint_S curl \, \vecs F \cdot dS\).

    18. \(\vecs F(x,y,z) = y\,\mathbf{\hat i} - x\,\mathbf{\hat j} + z\,\mathbf{\hat k} \), where \(S\) is the upper half of the unit sphere

    19. \(\vecs F(x,y,z) = y\,\mathbf{\hat i} + xyz \,\mathbf{\hat j} - 2zx\,\mathbf{\hat k} \), where \(S\) is the upward-facing paraboloid \(z = x^2 + y^2\) lying in cylinder \(x^2 + y^2 = 1\)

    Answer
    \(-\pi\)

     

    Use the divergence theorem to evaluate \(\iint_S \vecs F \cdot dS\).

    20. \(\vecs F(x,y,z) = (x^3y)\,\mathbf{\hat i} + (3y - e^x)\,\mathbf{\hat j} + (z + x)\,\mathbf{\hat k} \), over cube \(S\) defined by \(-1 \leq x \leq 1, \, 0 \leq y \leq 2, \, 0 \leq z \leq 2\)

    21. \(\vecs F(x,y,z) = (2xy)\,\mathbf{\hat i} + (-y^2)\,\mathbf{\hat j} + (2z^3)\,\mathbf{\hat k} \), where \(S\) is bounded by paraboloid \(z = x^2 + y^2\) and plane \(z = 2\)

    Answer
    \(31\pi /2\)

     

    22. Find the amount of work performed by a 50-kg woman ascending a helical staircase with radius 2 m and height 100 m. The woman completes five revolutions during the climb.

    23. Find the total mass of a thin wire in the shape of a semicircle with radius \(\sqrt{2}\), and a density function of \(\rho (x,y) = y + x^2\).

    Answer
    \(\sqrt{2}(2 + \pi)\)

    24. Find the total mass of a thin sheet in the shape of a hemisphere with radius \(2\) for \(z \geq 0\) with a density function \(\rho (x,y,z) = x + y + z\).

    25. Use the divergence theorem to compute the value of the flux integral over the unit sphere with \(\vecs F(x,y,z) = 3z\,\mathbf{\hat i} + 2y\,\mathbf{\hat j} + 2x\,\mathbf{\hat k} \).

    Answer
    \(2\pi /3\)

     

    Contributors

    Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.


    16R: Chapter 16 Review Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.