日: 2014年8月4日

TRANFORMATIONS OF MULTIPLE INTEGRALS

In evaluating a multiple integral over a region \cal R, it is often convenient to use coordinates other than rectangular, such as the curvilinear coordinates considered in Chapter 5.

If we let (u, v) be curvilinear coordinates of points in a plane, there will be a set of transformation equations x = f(u, v),\ y = g(u, v) mapping points (x, y) of the xy plane into points (u, v) of the uv plane. In such case the region \cal R of the xy plane is mapped into a region {\cal R}' of the uv plane. We then have

\displaystyle \underset{\cal R}{\iint}F(x, y)dxdy = \underset{{\cal R}'}{\iint} G(u, v)\left|\frac{\partial (x,y)}{\partial (u, v)}\right| dudv \cdots(9)

where G(u, v) \equiv F\{f(u,v), g(u,v)\} and

\displaystyle \frac{\partial (x, y)}{\partial (u, v)} \equiv \left| \begin{array}{cc} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{array} \right| \cdots (10)

is the Jacobian of x and y with respect to u and v.

Similarly if (u, v, w) are curvilinear coordinates in three dimensions, there will be a set of transformation equations x = f(u, v, w), y = g(u, v, w), z = h(u, v, w) and we can write

\displaystyle \underset{\cal R}{\iiint}F(x, y, z)dxdydz = \underset{{\cal R}'}{\iiint} G(u, v, w) \left| \frac{\partial (x, y, z)}{\partial (u, v, w)} \right|dudvdw \cdots(11)

where G(u, v, w) \equiv F\{f(u, v, w), g(u, v, w), h(u, v, w)\} and

\displaystyle \frac{\partial (x, y, z)}{\partial (u, v, w)} \equiv \left| \begin{array}{ccc} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w} \end{array} \right|\cdots(12)

is the Jacobian of x, y and z with respect to u, v and w.

The results (9) and (11) correspond to change of variables for double and triple integrals.

Generalizations to higher dimensions are easily made.

多重積分の変数変換

　領域 \cal R 上の多重積分を評価するにあたり，直交系以外の座標，例えば第5章で考慮したような曲線座標をしばしば便利に用いることがあります．

　仮に (u, v) をある平面上の点の曲線座標とすると，xy 平面の点 (x, y) を uv 平面の点 (u, v) にマッピングする変換式のセット x = f(u, v),\ y = g(u, v) が得られるはずです．その場合 xy 平面の領域 \cal R は uv 平面の領域 {\cal R}' にマッピングされます．ここで次の式を得ます．

\displaystyle \underset{\cal R}{\iint}F(x, y)dxdy = \underset{{\cal R}'}{\iint} G(u, v)\left|\frac{\partial (x,y)}{\partial (u, v)}\right| dudv \cdots(9)

ここで G(u, v) \equiv F\{f(u,v), g(u,v)\} および

\displaystyle \frac{\partial (x, y)}{\partial (u, v)} \equiv \left| \begin{array}{cc} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{array} \right| \cdots (10)

は u および v についての x および y の ヤコビアン です．

　同様に (u, v, w) を3次元における曲線座標とするなら x = f(u, v, w), y = g(u, v, w), z = h(u, v, w) という変数変換式セットが得られ，下記のように記述します．

\displaystyle \underset{\cal R}{\iiint}F(x, y, z)dxdydz = \underset{{\cal R}'}{\iiint} G(u, v, w) \left| \frac{\partial (x, y, z)}{\partial (u, v, w)} \right|dudvdw \cdots(11)

ここで G(u, v, w) \equiv F\{f(u, v, w), g(u, v, w), h(u, v, w)\} および

\displaystyle \frac{\partial (x, y, z)}{\partial (u, v, w)} \equiv \left| \begin{array}{ccc} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w} \end{array} \right|\cdots(12)

上記は u, v and w に関する x, y および z の ヤコビアン です．

　その結果 (9) および (11) は二重積分及び三重積分の変数変換に対応します．

　より高次への一般化も容易に可能です．