### The Calculus of Functions of Several Variables

Chapter 4

Section 4.1

2. The surface is the graph of $f$.

3. $F(s, t) = (s, t, s^2 + t^2)$

6. $\{\mathbf{x} \ : \ \mathbf{x} \in \mathbb{R}^n, \mathbf{x} \ne \mathbf{0}\}$

Section 4.2

1. (a) $\displaystyle{A(x,y) = \begin{bmatrix}2 & 4\\ 6 & 3\end{bmatrix} \begin{bmatrix}x - 1\\ y - 2\end{bmatrix} + \begin{bmatrix}5\\ 6\end{bmatrix} = \begin{bmatrix}2x + 4y - 5\\ 6x + 3y - 6\end{bmatrix}}$

(c) $\displaystyle{A(s,t) = \begin{bmatrix}-6 & \phantom{-}1\\ \phantom{-}1 & -1\\ 36 & -24\\ -1 & \phantom{-}4\end{bmatrix} \begin{bmatrix}s + 1\\ t - 3\end{bmatrix} + \begin{bmatrix}\phantom{-}6\\ -4\\ -36\\ \phantom{-}13\end{bmatrix} = \begin{bmatrix}-6s + t - 3\\ s - t\\ 36s - 24t + 72\\ -2 + 4t\end{bmatrix}}$

2. (a) $x = -2s + \pi$, $y = t$, $z = t$

(c) $\displaystyle{x = -\frac{1}{\sqrt{2}}\left(s - \frac{\pi}{2}\right)}$

$\displaystyle{y = \frac{1}{\sqrt{2}}\left(t - \frac{\pi}{4}\right) + \frac{1}{\sqrt{2}}}$

$\displaystyle{z = -\frac{1}{\sqrt{2}}\left(t - \frac{\pi}{2}\right) + \frac{1}{\sqrt{2}}}$

(e) $\displaystyle{x = -(2\sqrt{2} + 1)\left(s - \frac{3\pi}{4}\right) + \left(t - \frac{\pi}{4}\right) - 2\sqrt{2} - 1}$

$\displaystyle{y = -(2\sqrt{2} + 1)\left(s - \frac{3\pi}{4}\right) - \left(t - \frac{\pi}{4}\right) + 2\sqrt{2} + 1}$

$\displaystyle{y = \sqrt{2}\left(t - \frac{\pi}{4}\right) + \sqrt{2}}$

4. (a) $\displaystyle{D(f \circ g)(1, -2) = \begin{bmatrix}720 & -468\\ -8 & 15\end{bmatrix}}$

(c) $\displaystyle{D(f \circ g)(1, -2, 3) = \begin{bmatrix}92 & -89 & 3\\ 10920 & -9880 & 1040\\ -16 & 17 & 1\end{bmatrix}}$

6. $\displaystyle{\frac{\partial x}{\partial s} = (2uv)(-2s) + (u^2)\left(-\frac{4t}{s^2}\right)}$

$\displaystyle{\frac{\partial x}{\partial t} = (2uv)(8t) + (u^2)\left(\frac{4}{s}\right)}$

$\displaystyle{\frac{\partial y}{\partial s} = (3)(-2s) + (-1)\left(-\frac{4t}{s^2}\right)}$

$\displaystyle{\frac{\partial y}{\partial t} = (3)(8t) + (-1)\left(\frac{4}{s}\right)}$

8. (a) $\displaystyle{\left. \frac{\partial T}{\partial r}\right|_{r=4, \theta=\frac{\pi}{6}} = \frac{80}{17\sqrt{17}}}$, $\displaystyle{\left. \frac{\partial T}{\partial \theta}\right|_{r=4, \theta=\frac{\pi}{6}} = 0}$

(b) The level curves of $T$ are circles.

Section 4.3

1. (a) $\displaystyle{\int_C F \cdot ds = \frac{104}{5}}$

(c) $\displaystyle{\int_C F \cdot ds = -\frac{384}{5}}$

2. (a) $\displaystyle{\int_C 3xdx + 4ydy = 0}$

3. (a) $\displaystyle{\int_C 3xdx + 4ydy + zdz = 2\pi^2}$

4. (a) $\displaystyle{\int_C x^2ydx + (3y+x)dy = 0}$

5. $\displaystyle{\int_C F \cdot ds = -6\pi}$

6. (a) $\displaystyle{\int_C 3ydx = -\frac{27\pi}{2}}$

7. $\displaystyle{\int_C \frac{x}{x^2+y^2}dx + \frac{y}{x^2+y^2}dy = \frac{1}{2}\log(13)}$

Section 4.4

1. (a) $\displaystyle{\int_{\partial D} 2xydx + 3x^2dy = 80}$

2. (a) $\displaystyle{\int_{\partial D} 2xy^2dx + 4xdy = \frac{16}{3}}$ (c) $\displaystyle{\int_{\partial D} ydx - xdy = -8}$

4. $\displaystyle{\frac{3}{8}\pi a^2}$

5. $\displaystyle{\frac{\pi}{8}}$

6. $\displaystyle{\frac{9\pi}{2}}$