### The Calculus of Functions of Several Variables

Chapter 1

Section 1.1

2. (a) $(4, 5, 0)$

(b) $(12, 5, 4)$

(c) $(-12, 8, -{8})$

(d) $(-11, 7, -5)$

5. (a) $\sqrt{14}$

(b) $\sqrt{118}$

(c) $5\sqrt{14}$

(d) $3\sqrt{6}$

6. (a) $\sqrt{17}$

(b) $\sqrt{22}$

(c) $2\sqrt{3}$

(d) $\sqrt{66}$

(e) $\sqrt{19}$

11. No

12. (a) $||\mathbf{x}|| = \sqrt{5}$, Direction: $\displaystyle{||\mathbf{u}|| = \frac{1}{\sqrt{5}}(2, 1)}$

(b) $||\mathbf{z}|| = \sqrt{3}$, Direction: $\displaystyle{||\mathbf{u}|| = \frac{1}{\sqrt{3}}(1, 1, -1)}$

(c) $||\mathbf{x}|| = \sqrt{14}$, Direction: $\displaystyle{||\mathbf{u}|| = \frac{1}{\sqrt{14}}(-1, 2, 3)}$

(d) $||\mathbf{w}|| = \sqrt{15}$, Direction: $\displaystyle{||\mathbf{u}|| = \frac{1}{\sqrt{15}}(1, -1, 2, -3)}$

15. (a) $a = \dfrac{3}{2}$, $b = -\dfrac{1}{2}$; Yes, $a$ and $b$ are unique.

(b) $a = \dfrac{x + y}{2}$, $b = \dfrac{y - x}{2}$

Section 1.2

1. (a) $-16$

(b) $-32$

(c) $-58$

(d) $5$

3. (a) $-18$

(b) $-36$

(c) $-40$

(d) $-126$

4. (a) $0.6435$ radians, or $36.87^\circ$

(c) $1.9106$ radians, or $109.47^\circ$

(e) $0.6435$ radians, or $36.87^\circ$

5. The angle at vertex $(-2, 1)$ is $0.3218$ radians, at vertex $(1, 2)$ is $2.0344$ radians, and at vertex $(2, 1)$ is $\frac{\pi}{4}$ radians.

7. (a) $2.1588$ radians; $0.5880$ radians

(b) $1.9913$ radians; $0.6155$ radians; $1.1503$ radians

(c) $1.0282$ radians; $0.6847$ radians; $1.3096$ radians; $1.8320$ radians

(d) $1.4355$ radians; $1.2977$ radians; $1.1543$ radians; $1.011$ radians; $0.8309$ radians

8. (a) Coordinate: $-\dfrac{4}{\sqrt{17}}$; Projection: $\displaystyle{\left(-\frac{16}{17},-\frac{4}{17}\right)}$

(b) Coordinate: $\dfrac{3}{\sqrt{11}}$; Projection: $\displaystyle{\left(-\frac{3}{11},\frac{9}{11},\frac{3}{11}\right)}$

(c) Coordinate: $\dfrac{5}{\sqrt{38}}$; Projection: $\displaystyle{\left(\frac{5}{38},-\frac{5}{38},\frac{15}{19}\right)}$

(d) Coordinate: $\dfrac{5}{3\sqrt{2}}$; Projection: $\displaystyle{\left(\frac{5}{9},-\frac{5}{18},\frac{5}{9},\frac{5}{6}\right)}$

11. $\displaystyle{\mathbf{x} = \left(\frac{5}{7},\frac{15}{14},\frac{5}{14}\right) + \left(\frac{2}{7}, \frac{13}{14},-\frac{47}{14}\right)}$

Section 1.3

1. (a) $\mathbf{x} \times \mathbf{y} = (1, 3, 7)$

(b) $\mathbf{x} \times \mathbf{y} = (-2, 16, -5)$

(c) $\mathbf{x} \times \mathbf{y} = (36, -12, 0)$

(d) $\mathbf{x} \times \mathbf{y} = (-6, 2, -14)$

3. $11$

5. $3$

7. $\dfrac{9}{2}$

9. $42$

13. For example, $\mathbf{e}_2 \times \left(\mathbf{e}_2 \times \mathbf{e}_3\right) = -\mathbf{e}_3$, whereas $\left(\mathbf{e}_2 \times \mathbf{e}_2\right) \times \mathbf{e}_3 = \mathbf{0}$.

Section 1.4

1. Vector equation: $\mathbf{y} = t(1, -2) + (2, 3) = (t + 2, -2t + 3)$

Parametric equations: \begin{align} x &= t +2 \\ y &= -2t + 3 \end{align}

3. (a) Vector equation: $\mathbf{y} = t(5,5) + (-1, -3) = (5t - 1, 5t - 3)$

Parametric equations: \begin{align} x &= 5t -1 \\ y &= 5t -3 \end{align}

(b) Vector equation:

$\mathbf{y} = t(3, 1, -2) + (2, 1, 3) = (3t+2, t+1, -2t+3)$

Parametric equations: \begin{align} x &= 3t + 2 \\ y &= t + 1 \\ z &= -2t + 3 \end{align}

(c) Vector equation:

$\mathbf{y} = t(1, 2, -3, 3) + (3, 2, 1, 4) = (t+3, 2t+2,-3t+1,3t+4)$

Parametric equations: \begin{align} w &= t + 3 \\ x &= 2t + 2 \\ y &= -3t + 1 \\ z &= 3t + 4 \end{align}

(d) Vector equation:

$\mathbf{y} = t(3, -1, -2) + (4, -3, 2) = (3t+4, -t-3, -2t+2)$

Parametric equations: \begin{align} x &= 3t + 4 \\ y &= -t - 3 \\ z &= -2t + 2 \end{align}

5. $\dfrac{1085}{7}$

7. $\dfrac{4697}{14}$

9. Vector equation:

$\mathbf{y} = t(-3, 6, -2, -3) + s(0, 2, -2, 2) + (2, 3, 4, -1)$

Parametic equations: \begin{align} w &= -3t + 2 \\ x &= 6t + 2s + 3 \\ y &= -2t - 2s + 4 \\ z &= -3t + 2s - 1 \end{align}

11. $3$

13. $\mathbf{n} = (1, 0)$

Normal equation: $(1, 0) \cdot (x-2, y) = 0$, or $x = 2$

15. $\mathbf{n} = (1, 4)$

Normal equation: $(1, 4) \cdot (x-3, y-2) = 0$, or $x + 4y = 11$

16. $\mathbf{n} = (11, 8, 7)$

Normal equation:

$(11, 8, 7) \cdot (x-1, y-2, z+1) = 0$, or $11x + 8y + 7z = 20$

17. $\dfrac{4}{\sqrt{5}}$

19. $\dfrac{3}{\sqrt{23}}$

21. $0.7017$ radians

23. $2x - y = 3$ is the equation of one such plane.

27. $y = 2t - \frac{2}{3}$, $z = -s - t + \frac{11}{3}$

Section 1.5

2. (a) Dimension of the domain space $= 2$; dimension of the range space $= 3$; $f$ is linear

(b) Dimension of the domain space $= 2$; dimension of the range space $= 2$, $f$ is neither linear nor affine

(c) Dimension of the domain space $= 3$; dimension of the range space $= 3$; $f$ is linear

(d) Dimension of the domain space $= 3$; dimension of the range space $= 2$; $f$ is linear

(e) Dimension of the domain space $= 3$; dimension of the range space $= 4$; $f$ is affine

(f) Dimension of the domain space $= 2$; dimension of the range space $= 1$; $f$ is affine

(g) Dimension of the domain space $= 1$; dimension of the range space $= 2$; $f$ is linear

(h) Dimension of the domain space $= 4$; dimension of the range space $= 2$; $f$ is linear

(i) Dimension of the domain space $= 2$; dimension of the range space $= 2$; $f$ is neither linear nor affine

(j) Dimension of the domain space $= 2$; dimension of the range space $= 3$; $f$ is neither linear nor affine

3. (a) $\displaystyle{M = \begin{bmatrix}1 & \phantom{-}1\\ 2 & -3\end{bmatrix}}$

(b) $\displaystyle{M = \begin{bmatrix}2 & 1 & -1 & \phantom{-}3\\ 1 & 2 & \phantom{-}0 & -3 \end{bmatrix}}$

(c) $\displaystyle{M = \begin{bmatrix}3 \\ 1 \\ 4\end{bmatrix}}$

(d) $\displaystyle{M = \begin{bmatrix}-5\end{bmatrix}}$

(e) $\displaystyle{M = \begin{bmatrix}4 & -3 & 2\end{bmatrix}}$

(f) $\displaystyle{M = \begin{bmatrix}1 & \phantom{-}1 & 1\\ 3 & -1 & 0\\ 0 & \phantom{-}1 & 2\end{bmatrix}}$

(g) $\displaystyle{M = \begin{bmatrix}2 & \phantom{-}0\\ 0 & \phantom{-}3\\ 1 & \phantom{-}1\\ 1 & -1\\ 2 & -3\end{bmatrix}}$

(h) $\displaystyle{M = \begin{bmatrix}1 & 0\\ 0 & 1\end{bmatrix}}$

(i) $\displaystyle{M = \begin{bmatrix}2 & 1 & -1 & \phantom{-}3\\ 1 & 2 & \phantom{-}0 & -3\end{bmatrix}}$

5. (a) $\displaystyle{\begin{bmatrix}-4 \\ \phantom{-}4\end{bmatrix}}$

(b) $\displaystyle{\begin{bmatrix}-5\\ 11\\ -4\end{bmatrix}}$

(c) $\displaystyle{\begin{bmatrix}3\end{bmatrix}}$

(d) $\displaystyle{\begin{bmatrix}2 \\ 10\\ 3\end{bmatrix}}$

7. $\displaystyle{M = \begin{bmatrix}0 & 1\\ 1 & 0\end{bmatrix}}$

8. $\displaystyle{M = \begin{bmatrix}\phantom{-}0 & -1\\ -1 & \phantom{-}0\end{bmatrix}}$

10. $\displaystyle{M = \begin{bmatrix}\phantom{-}\cos(\theta) & \phantom{-}\sin(\theta)\\ -\sin(\theta) & \phantom{-}\cos(\theta)\end{bmatrix}}$

Section 1.6

1. (a) $\displaystyle{\begin{bmatrix}\phantom{-}6 & \phantom{-}9\\ -6 & \phantom{-}3\\ 12 & -3\end{bmatrix}}$

(b) $\displaystyle{\begin{bmatrix}-1 & 5\\ -3 & 1\\ \phantom{-}2 & 4\end{bmatrix}}$

(c) $\displaystyle{\begin{bmatrix}\phantom{-}7 & \phantom{-}4\\ -3 & \phantom{-}2\\ 10 & -7\end{bmatrix}}$

(d) $\displaystyle{\begin{bmatrix}-14 & 18\\ -10 & 2\\ -4 & 28\end{bmatrix}}$

2. (a) $\displaystyle{\begin{bmatrix}12\\ 1\end{bmatrix}}$

(b) $\displaystyle{\begin{bmatrix}-4 & 14\\ \phantom{-}9 & -4\end{bmatrix}}$

(c) $\displaystyle{\begin{bmatrix}12 & 13 & -4\\ -7 & -7 & \phantom{-}9\end{bmatrix}}$

(d) $\displaystyle{\begin{bmatrix}2 & 19\end{bmatrix}}$

3. (a) $\displaystyle{\begin{bmatrix}\phantom{-}6 & 9 & \phantom{-}0\\ -3 & 3 & \phantom{-}6\\ \phantom{-}3 & 6 & -3\end{bmatrix}}$

(b) $\displaystyle{\begin{bmatrix}4 & 7 & -3\\ 0 & 2 & \phantom{-}3\\ 4 & 1 & \phantom{-}3\end{bmatrix}}$

(c) $\displaystyle{\begin{bmatrix}\phantom{-}2 & 2 & \phantom{-}3\\ -3 & 1 & \phantom{-}3\\ -1 & 5 & -6\end{bmatrix}}$

(d) $\displaystyle{\begin{bmatrix}\phantom{-}6 & 10 & -3\\ -1 & 3 & \phantom{-}5\\ \phantom{-}5 & 3 & \phantom{-}2\end{bmatrix}}$

(e) $\displaystyle{\begin{bmatrix}-3 & 4 & 11\\ 2 & 6 & 1\\ 11 & 16 & -6\end{bmatrix}}$

(f) $\displaystyle{\begin{bmatrix}7 & 11 & -3\\ 5 & -5 & 12\\1 & 7 & -5\end{bmatrix}}$

5. (a) $5$

(b) $-4$

(c) $175$

(d) $17$

(e) $-143$

(f) $300$

7. $32$

8. $8$

14. This is the set of all points which satisfy $x - y - z = 0$.