과제 1

기한: 2015-03-20 17:00

연습문제 1-4.
Show
(a) $( \boldsymbol{\mathsf{A}} \boldsymbol{\mathsf{B}} )^t = {\boldsymbol{\mathsf{B}}}^t {\boldsymbol{\mathsf{A}}}^t$
(b) $( \boldsymbol{\mathsf{A}} \boldsymbol{\mathsf{B}} )^{-1} = {\boldsymbol{\mathsf{B}}}^{-1} {\boldsymbol{\mathsf{A}}}^{-1}$

연습문제 1-11.
Show that the triple scalar product $( \vec{A} \times \vec{B} ) \cdot \vec{C}$ can be written as

(1)
\begin{align} ( \vec{A} \times \vec{B} ) \cdot \vec{C} = \begin{vmatrix} A_1 & A_2 & A_3 \\ B_1 & B_2 & B_3 \\ C_1 & C_2 & C_3 \end{vmatrix} \end{align}

Show also that the product is unaffected by an interchange of the scalar and vector product operations or by a change in the order of $\vec{A}, \vec{B}, \vec{C}$, as long as they are in cyclic order; that is,

(2)
\begin{align} ( \vec{A} \times \vec{B} ) \cdot \vec{C} = \vec{A} \cdot ( \vec{B} \times \vec{C} ) = \vec{B} \cdot ( \vec{C} \times \vec{A} ) = ( \vec{C} \times \vec{A} ) \cdot \vec{B}, \qquad \rm{etc.} \end{align}

We may therefore use the notation $\vec{A}\vec{B}\vec{C}$ to denote the triple scalar product. Finally, give a geometric interpretation of $\vec{A}\vec{B}\vec{C}$ by computing the volume of the parallelepiped defined by the three vectors $\vec{A}, \vec{B}, \vec{C}$.

연습문제 1-21.
Show (see also Problem 1-11) that

(3)
\begin{align} \vec{A}\vec{B}\vec{C} = \sum_{i,j,k} \epsilon_{ijk}\ A_i B_j C_k \end{align}

연습문제 1-23.
Use the $\epsilon_{ijk}$ notation and derive the indentity

(4)
\begin{align} ( \vec{A} \times \vec{B} ) \times ( \vec{C} \times \vec{D} ) = ( \vec{A} \vec{B} \vec{C} ) \vec{C} - ( \vec{A} \vec{B} \vec{C}) \vec{D} \end{align}

연습문제 1-26
A particle moves with $v = \rm{const.}$ along the curve $r = k(1 + \cos{\theta} )\ ( \mathrm{a}\ cardioid )$. Find $\ddot{\vec{r}} \cdot \vec{e}_r = \vec{a} \cdot {\vec{e}}_r$, $\left| \vec{a} \right|$, and $\dot{\theta}$

연습문제 1-27.
If $\vec{r}$ and $\dot{\vec{r}} = \vec{v}$ are both explicit functions of time, show that

(5)
\begin{align} { d \over {dt}} \left[ \vec{r} \times ( \vec{v} \times \vec{r} ) \right] = r^2 \vec{a} + ( \vec{r} \cdot \vec{v} ) \vec{v} - (v^2 + \vec{r} \cdot \vec{a}) \vec{r} \end{align}

연습문제 1-30.
Show that

(6)
\begin{align} \vec{\nabla} (\phi \psi ) = \phi \vec{\nabla} \psi + \psi \vec{\nabla} \phi \end{align}

연습문제 1-32.
Show that

(7)
\begin{align} \int ( 2a \vec{r} \cdot \dot{\vec{r}} + 2 b \dot{\vec{r}} \cdot \ddot{\vec{r}} ) dt = a {\vec{r}}^2 + b {\dot{r}}^2 + \mathrm{const.} \end{align}

연습문제 1-36.
Find the value of the integral $\int_{S} \vec{A} \cdot d \vec{a}$, where $\vec{A} = x \vec{i} - y \vec{j} + z \vec{k}$ and $S$ is the closed surface defined by the cylinder $c^2 = x^2 + y^2$. The top and bottom of the cylinder are at $z = d$ and $0$, respectively.

연습문제 1-38.
Find the value of the integral $\int_{S} ( \vec{\nabla} \times \vec{A} ) \cdot d \vec{a}$ if the vector $\vec{A} = y \vec{i} + z \vec{j} + x \vec{k}$ and $S$ is the surface defined by the paraboloid $z = 1 - x^2 - y^2$, where $z \geq 0$.