
# 16R: 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.

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.

False

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

Draw the following vector fields.

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

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

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

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

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

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 $$(4, 2)$$

$$-\dfrac{16}{3}$$

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 $$z = x^2 - y, \, 0 \leq x \leq 1, \, 0 \leq y \leq 4$$

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

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

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 $$3.$$

$$-2\pi$$

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 $$x^2 + y^2 = 1$$

$$-\pi$$

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 $$z = 2$$

$$31\pi /2$$

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 $$\rho (x,y) = y + x^2$$.

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

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 $$\vecs F(x,y,z) = 3z\,\mathbf{\hat i} + 2y\,\mathbf{\hat j} + 2x\,\mathbf{\hat k}$$.

$$2\pi /3$$