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Appendix
 
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Problems

A.3
For the following vectors calculate $\vert\mathbf{a}\vert$, $\mathbf{a}\cdot\mathbf{b}$, $\mathbf{a}\times\mathbf{b}$ and $\mathbf{a}\otimes\mathbf{b}$. Show the calculations by writing $\mathbf{a}$ and $\mathbf{b}$ in matrix form.

\begin{eqnarray*} \text{a)}&\mathbf{a}=-3\mathbf{i}+4\mathbf{k}&\mathbf{b}=2\mat... ...z& \mathbf{b}=13\mathbf{e}_r-2\mathbf{e}_{\theta}+5\mathbf{e}_z \end{eqnarray*}



A.4
Consider the vectors $\mathbf{a}=2\mathbf{i}-5\mathbf{j}-2\mathbf{k}$ and $\mathbf{b}=\mathbf{i}-3\mathbf{j}$
a)
Resolve $\mathbf{a}$ into two vectors: one in the direction of $\mathbf{b}$ and the other normal to $\mathbf{b}$.
b)
Sketch $\mathbf{a}$, $\mathbf{b}$ and the two components of $\mathbf{a}$ found above.
c)
Determine the dot product of the two components of $\mathbf{a}$.
d)
Find a vector $\mathbf{c}$ that is perpendicular to both $\mathbf{a}$ and $\mathbf{b}$.
A.5
For the following questions use the vectors $\mathbf{a}=7\mathbf{i}+3\mathbf{j}-2\mathbf{k}$, $\mathbf{b}=2\mathbf{i}+\mathbf{j}+3\mathbf{k}$, and $\mathbf{c}=\mathbf{i}-4\mathbf{j}+\mathbf{k}$. Find:
a.)
$\mathbf{a}\cdot\mathbf{b}$
b.)
$\mathbf{b}\times\mathbf{c}$
c.)
$\mathbf{c}\otimes\mathbf{b}$
d.)
$\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})$
e.)
$(\mathbf{a}\times\mathbf{c})\times(\mathbf{b}\times\mathbf{c})$
f.)
$\mathbf{a}\times(\mathbf{b}+\mathbf{c})$
g.)
$(\mathbf{a}\times\mathbf{c})+(\mathbf{b}\times\mathbf{c})$
A.6
For the velocity vectors $\mathbf{v}_1=3xy\mathbf{i}-2y\mathbf{j}+xyz\mathbf{k}$, $\mathbf{v}_2=(x^2-3x)\mathbf{i}+(y^3+2y-7)\mathbf{j}+z\mathbf{k}$ and $\mathbf{v}_3=(x^22y^2-5z)\mathbf{i}+(y^3+2z)\mathbf{j}+(7yz)\mathbf{k}$, determine the following:
a)
The divergence of $\mathbf{v}$: $\nabla\cdot\mathbf{v}$
b)
The curl of $\mathbf{v}$: $\nabla\times\mathbf{v}$
c)
$\mathbf{v}\cdot(\nabla\mathbf{v})$
d)
$(\mathbf{v}\cdot\nabla)\mathbf{v}$
A.7
For the vector-valued functions of position $\mathbf{u}$ and $\mathbf{v}$ (velocities) and for the scalar-valued function of position $\rho$ (density), prove the following identities. Use a rectangular coordinate representation for $\nabla$, $\mathbf{u}$ and $\mathbf{v}$ (e.g., $\mathbf{u}=u_x\mathbf{i}+u_y\mathbf{j}+u_z\mathbf{k}$) to compute the expressions on each side of each equation. Verify using Scientific Workplace.
a)
$\nabla\cdot\left(\rho\mathbf{v}\right)=\mathbf{v}\cdot\nabla\rho+\rho\nabla\cdot\mathbf{v}$
b)
$\nabla\times\left(\rho\mathbf{v}\right)=\nabla\rho\times\mathbf{v}+\rho\nabla\times\mathbf{v}$
c)
$\nabla\cdot\mathbf{u}\times\mathbf{v})=(\nabla\times\mathbf{u})\cdot\mathbf{v} -\mathbf{u}\cdot\nabla\times\mathbf{v})$
d)
$\nabla\cdot(\nabla\times\mathbf{u})=0$
A.8
Show that $\nabla\times\nabla\Phi\equiv 0$, where $\Phi$ is a scalar-valued function of position, i.e., $\Phi=\Phi(\,x,y,z\,)$.
A.9
Given the matrices:

\begin{displaymath}\left[ A \right]=\left[\begin{array}{ccc} 5 & -3 & 4\\ 6 & 2 ... ...right]=\left[\begin{array}{c} 7\\ 5\\ -3\\ \end{array}\right], \end{displaymath}

evaluate the following:
(a)
$[B]+[D]$
(b)
$[A][A]^T$
(c)
$[A][B]$
(d)
$[C][A][B]$
(e)
$[C][D]$
(f)
$[D][C]$
A.10
Given the following set of linear equations in matrix form, write the equations as three separate equations.

\begin{displaymath}\left[\begin{array}{ccc} 5 & -3 & 4\\ 6 & 2 & -5\\ -6 & 3 & 4... ...\right]=\left[\begin{array}{c} 2\\ -1\\ 4\\ \end{array}\right] \end{displaymath}

A.11
Write the following linear algebraic equations in matrix form:

\begin{eqnarray*} 6C_1-2C_2&=&16\ -2C_1+4C_2+C_3&=&5\ C_2+8C_4&=&-7 \end{eqnarray*}



A.12
Solve for $\mathbf{x}$ from the set of equations

\begin{displaymath}\left[\begin{array}{ccc} 5 & -3 & 4\\ 6 & 2 & -5\\ -6 & 3 & 4... ...\right]=\left[\begin{array}{c} 2\\ -1\\ 4\\ \end{array}\right] \end{displaymath}


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