====The Calculus of Functions of Several Variables====

===Answers for selected problems===

**Chapter 1**

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**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} $

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**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)}$

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**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} $.

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**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} $

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**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}}$

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**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 $.

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