9E: Chapter Exercises

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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: \(2\pi /3\)