9E: Chapter Exercises

Exercise $\PageIndex{1}$: True or False?

Justify your answer with a proof or a counterexample.

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

Answer

False

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

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

Answer

False

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

Exercise $\PageIndex{2}$

Draw the following vector fields.

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

Answer

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

Exercise $\PageIndex{3}$

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

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

Answer

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

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

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

Answer

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

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

Exercise $\PageIndex{4}$

Evaluate the following integrals.

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

Answer

$-\dfrac{16}{3}$

2. $\int_C ydx + xy^2 dy$, where $C : x = \sqrt{t}, \, y = t - 1, \, 0 \leq t \leq 1$

3. $\iint_S xy^2 dS,$ where S is surface $z = x^2 - y, \, 0 \leq x \leq 1, \, 0 \leq y \leq 4$

Answer

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

Exercise $\PageIndex{5}$

Find the divergence and curl for the following vector fields.

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

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

Answer

Divergence: $e^x + x \, e^{xy} + xy\, e^{xyz}$, curl: $xz e^{xyz} \, \hat{ \mathbf i} - yz e^{xyz} \hat{j }+ ye^{xy} \hat{ k}$

Exercise $\PageIndex{6}$

Use Green’s theorem to evaluate the following integrals.

1. $\int_C 3xydx + 2xy^2 dy$, where $C$ is a square with vertices $(0, 0), (0, 2), (2, 2)$ and $(2, 0)$.

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

Answer

$-2\pi$

Exercise $\PageIndex{7}$

Use Stokes’ theorem to evaluate $\iint_S curl \, \vecs{F} \cdot \vecs{dS}$.

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

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

Answer

$-\pi$

Exercise $\PageIndex{8}$

Use the divergence theorem to evaluate $\iint_S\vecs{F} \cdot \vecs{dS}$.

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

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

Answer

$31\pi /2$

Exercise $\PageIndex{9}$

1. 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.

2. 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

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

3. 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 $\rho (x,y,z) = x + y + z$.

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

Answer

$2\pi /3$