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.