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Next: Coulombの法則 Up: denjiki Previous: denjiki

電磁気で使う数学

\fbox{\parbox{14.5cm}{
\begin{itemize}
\item 内積 (スカラー積)
\begin{display...
...l y} + \bm{e}_z\frac{\partial}{\partial z}
\end{displaymath}
\end{itemize}
}}

[ 例題1 ] ナブラ演算


$\displaystyle \bm{r}$ $\displaystyle =$ $\displaystyle (x, y, z)$  
$\displaystyle r$ $\displaystyle =$ $\displaystyle \sqrt{x^2 + y^2 + z^2} = t^{1/2}$  
$\displaystyle t$ $\displaystyle =$ $\displaystyle x^2 + y^2 + z^2$  
$\displaystyle \frac{\partial r}{\partial x}$ $\displaystyle =$ $\displaystyle \frac{t^{-1/2}}2 2x = \frac{x}{\sqrt{x^2 + y^2 + z^2}} = \frac{x}{r}$  
$\displaystyle \nabla r$ $\displaystyle =$ $\displaystyle \left( \frac{x}{r}, \frac{y}{r}, \frac{z}{r} \right) = \frac{\bm{r}}{r} = \bm{e}_r$  

\fbox{\parbox{14.5cm}{
\begin{itemize}
\item Taylor展開
\par
$x = a$のまわりで...
...) + xf'(0) + \frac{x^2}{2} f''(0) + \cdots
\end{displaymath}
\end{itemize}
}}

[ 例題2 ] 関数の展開

$ x=0$ のまわりで

$\displaystyle \sin x$ $\displaystyle =$ $\displaystyle 0 + x + 0 - \frac{x^3}{6} + \cdots$  
$\displaystyle \cos x$ $\displaystyle =$ $\displaystyle 1 - 0 - \frac{x^2}{2} + 0 + \cdots$  
$\displaystyle (1+x)^{\alpha}$ $\displaystyle =$ $\displaystyle 1 + \alpha x + \alpha(\alpha-1) \frac{x^2}{2} + \cdots$  

$ x=1$ のまわりで

$\displaystyle \ln x = 0 + (x-1) - \frac{(x-1)^2}{2} + \frac{(x-1)^3}{3} + \cdots \\
$

[ 例題3 ] 3次元関数の展開


$\displaystyle \bm{r}$ $\displaystyle =$ $\displaystyle (x, y, z)$  
$\displaystyle \bm{a}$ $\displaystyle =$ $\displaystyle (a_x, a_y, a_z)$  

\begin{displaymath}
\left\{
\begin{array}{ccl}
\Delta x &=& x - a_x \\
\Delta y &=& y - a_y \\
\Delta z &=& z - a_z
\end{array}
\right.
\end{displaymath}


$\displaystyle f(\bm{r})$ $\displaystyle =$ $\displaystyle f(\bm{a}) + \left( \left. \Delta x \frac{\partial f}{\partial x} ...
...ft. \Delta z \frac{\partial f}{\partial z} \right\vert _{\bm{r}=\bm{a}} \right)$  
    $\displaystyle + \frac{1}{2} \left( \left. (\Delta x)^2 \frac{\partial^2 f}{\par...
...lta z)^2 \frac{\partial^2 f}{\partial z^2} \right\vert _{\bm{r}=\bm{a}} \right.$  
    $\displaystyle + \left. \left. 2\Delta x\Delta y \frac{\partial^2 f}{\partial x\...
... \frac{\partial^2 f}{\partial z\partial x} \right\vert _{\bm{r}=\bm{a}} \right)$  
  $\displaystyle =$ $\displaystyle f(\bm{a}) + (\bm{r}-\bm{a})\cdot\nabla f(\bm{a}) + \frac{\{(\bm{r}-\bm{a})\cdot\nabla\}^2}{2} f(\bm{a}) + \cdots$  

\fbox{\parbox{14.5cm}{
\begin{itemize}
\item 置換積分
\begin{displaymath}
\i...
...gin{displaymath}
\int u'v = uv - \int uv'
\end{displaymath}
\end{itemize}
}}

[ 例題4 ] 積分


$\displaystyle I$ $\displaystyle =$ $\displaystyle \int \frac{dx}{x^2 + a^2}$  
    $\displaystyle {\color{blue} x = at とおくと dx = adt}$  
  $\displaystyle =$ $\displaystyle \int \frac{adt}{a^2t^2 + a^2}$  
  $\displaystyle =$ $\displaystyle \frac{1}{a} \int \frac{dt}{t^2 + 1}$  
    $\displaystyle {\color{blue} t = \tan\theta とおくと t^2 + 1 = \frac{1}{\cos^2\theta}}$  
    $\displaystyle {\color{blue} dt = \frac{d\theta}{\cos^2\theta}}$  
  $\displaystyle =$ $\displaystyle \frac{1}{a} \int \frac{\cos^2 \theta}{\cos^2 \theta}d\theta$  
  $\displaystyle =$ $\displaystyle \frac{1}{a} \int d\theta$  
  $\displaystyle =$ $\displaystyle \frac{1}{a} (\theta + C)$  
       
$\displaystyle \int_{-\infty}^{\infty} \frac{dx}{x^2 + a^2}$ $\displaystyle =$ $\displaystyle \frac{1}{a} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} d\theta
= \frac{\pi}{a}$  

\fbox{\parbox{14.5cm}{
\begin{itemize}
\item 線積分
\par
ベクトル$\bm{A}$を経弄..
...splaymath}
I = \int_C \bm{A}\cdot d\bm{s}
\end{displaymath}
\end{itemize}
}}

[ 例題5 ] 線積分

\includegraphics[width=3.5cm]{line-int.eps}

$\displaystyle \bm{A}$ $\displaystyle =$ $\displaystyle (x^2+y, xy, 0)$  
$\displaystyle d\bm{s}$ $\displaystyle =$ $\displaystyle (dx, dy, 0)$  
$\displaystyle \bm{A}\cdot d\bm{s}$ $\displaystyle =$ $\displaystyle (x^2+y)dx + xydy$  


$\displaystyle \int_C \bm{A}\cdot d\bm{s}$ $\displaystyle =$ $\displaystyle \int_0^a A_x(y=0)dx + \int_0^a A_y(x=a)dy + \int_a^0 A_x(y=a)dx + \int_a^0 A_y(x=0)dy$  
  $\displaystyle =$ $\displaystyle \int_0^a x^2dx + \int_0^a aydy + \int_a^0 (x^2+a)dx + \int_a^0 0dy$  
  $\displaystyle =$ $\displaystyle \left[ \frac{x^3}{3} \right]_0^a + \left[ \frac{ay^2}{2} \right]_0^a + \left[ \frac{x^3}{3} + ax \right]_a^0$  
  $\displaystyle =$ $\displaystyle \frac{a^3}{3} + \frac{a^3}{2} - \frac{a^3}{3} - a^2$  
  $\displaystyle =$ $\displaystyle \frac{a^3}{2} - a^2$  

\fbox{\parbox{14.5cm}{
\begin{itemize}
\item 面積分
\begin{displaymath}
I = ...
...cm}{$d\bm{S} = a^2\sin\theta d\varphi\bm{e}_r$}
\end{center}
\end{itemize}
}}

[ 例題6 ] 面積分

\includegraphics[width=3.5cm]{area-int.eps}

$\displaystyle \bm{A}$ $\displaystyle =$ $\displaystyle (x^2+y, xy, 0)$  
$\displaystyle \nabla\times\bm{A}$ $\displaystyle =$ $\displaystyle (y-1)\bm{e}_z$  
$\displaystyle d\bm{S}$ $\displaystyle =$ $\displaystyle dxdy\bm{e}_z$  


$\displaystyle \int \nabla\times\bm{A}\cdot d\bm{S}$ $\displaystyle =$ $\displaystyle \int_0^a dx \int_0^a (y-1)dy$  
  $\displaystyle =$ $\displaystyle a \left[ \frac{y^2}{2} - y \right]_0^a$  
  $\displaystyle =$ $\displaystyle \frac{a^3}{2} - a^2$  

\fbox{\parbox{14.5cm}{
\begin{itemize}
\item 体積分
\begin{displaymath}
I = ...
...}
\begin{displaymath}
dV = rdrd\theta dz
\end{displaymath}
\end{itemize}
}}

[ 例題7 ] 球の体積

\includegraphics[width=3cm]{sphere.eps}

$\displaystyle V$ $\displaystyle =$ $\displaystyle \int_0^a r^2dr \int_0^{\pi} \sin\theta d\theta \int_0^{2\pi} d\varphi$  
  $\displaystyle =$ $\displaystyle \left[ \frac{r^3}{3} \right]_0^a \left[ -\cos\theta \right]_0^{\pi} 2\pi$  
  $\displaystyle =$ $\displaystyle \frac{4\pi a^3}{3}$  


next up previous
Next: Coulombの法則 Up: denjiki Previous: denjiki
Keiichi Takasugi
平成24年1月25日