16.R: Chapter 16 Review Exercises

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

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.