
9E: Chapter Exercises


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

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

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.

False

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

Exercise $$\PageIndex{2}$$

Draw the following vector fields.

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

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

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

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

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

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)

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

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

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

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  $$3$$.

$$-2\pi$$

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

$$-\pi$$

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 $$z = 2$$

$$31\pi /2$$

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

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