重積分、ヤコビアン
１ 領域$${D=\{(x,y)| a\leqq x\leqq b, c\leqq y\leqq d\}}$$のとき
x軸方向に積分してからy軸方向に積分するか、その逆か
$${\displaystyle\int\displaystyle\int_Df(x,y)dxdy=\displaystyle\int_a^b\Big\{\displaystyle\int_c^df(x,y)dy\Big\}dx= …}$$
または $${ =\displaystyle\int_c^d\Big\{\displaystyle\int_a^bf(x,y)dx\Big\}dy= …}$$
２ 領域$${D=\{(x,y)| a\leqq x\leqq b, g_1(x)\leqq y\leqq g_2(x)\}}$$のとき
$${\displaystyle\int\displaystyle\int_Df(x,y)dxdy=\displaystyle\int_a^b\Big\{\displaystyle\int_{g_1(x)}^{g_2(x)}f(x,y)dy\Big\}dx= …}$$
３ 領域$${D=\{(x,y)| h_1(y)\leqq x\leqq h_2(y), c\leqq y\leqq d\}}$$のとき
$${\displaystyle\int\displaystyle\int_Df(x,y)dxdy=\displaystyle\int_c^d\Big\{\displaystyle\int_{h_1(y)}^{h_2(y)}f(x,y)dx\Big\}dy= …}$$
４ 領域$${D=\{(x,y)| a\leqq u(x,y)\leqq b, c\leqq v(x,y)\leqq d\}}$$のとき
$${\displaystyle\int\displaystyle\int_Df(x,y)dxdy}$$を$${\displaystyle\int\displaystyle\int_{D'}g(u,v)dudv=\displaystyle\int_a^b\displaystyle\int_c^dg(u,v)dudv}$$
の形に変形すればよい。
$${dx,dy}$$を$${du,dv}$$で表わす。
$${dx=\dfrac{\partial x}{\partial u}du+\dfrac{\partial x}{\partial v}dv}$$、 $${dy=\dfrac{\partial y}{\partial u}du+\dfrac{\partial y}{\partial v}dv}$$
よって
$${\begin{pmatrix}dx\\\\dy\end{pmatrix}=\begin{pmatrix}\dfrac{\partial x}{\partial u}du & \dfrac{\partial x}{\partial v}dv\\ \\ \dfrac{\partial y}{\partial u}du & \dfrac{\partial y}{\partial v}dv \end{pmatrix}=\begin{pmatrix}\dfrac{\partial x}{\partial u} & \dfrac{\partial x}{\partial v}\\\\ \dfrac{\partial y}{\partial u} & \dfrac{\partial y}{\partial v} \end{pmatrix}\begin{pmatrix}du\\\\dv\end{pmatrix}}$$
座標系O$${d_ud_v}$$においてＰ($${d_u,0}$$)､Ｑ($${0,d_v}$$)、座標系O$${dxdy}$$においてＰ'($${dx_P,dy_P}$$)､Ｑ($${dx_Q,dy_Q}$$)とする。
上式にＰ､Ｑの座標を入れＰ’､Ｑ'の座標を求める。
$${\begin{pmatrix}dx_P\\\\dy_P\end{pmatrix}=\begin{pmatrix}\dfrac{\partial x}{\partial u} & \dfrac{\partial x}{\partial v}\\\\ \dfrac{\partial y}{\partial u} & \dfrac{\partial y}{\partial v} \end{pmatrix}\begin{pmatrix}du\\\\0\end{pmatrix}=\begin{pmatrix}\dfrac{\partial x}{\partial u}du\\\\ \dfrac{\partial y}{\partial u}du\end{pmatrix}}$$
$${\begin{pmatrix}dx_Q\\\\dy_Q\end{pmatrix}=\begin{pmatrix}\dfrac{\partial x}{\partial u} & \dfrac{\partial x}{\partial v}\\\\ \dfrac{\partial y}{\partial u} & \dfrac{\partial y}{\partial v} \end{pmatrix}\begin{pmatrix}0\\\\dv\end{pmatrix}=\begin{pmatrix}\dfrac{\partial x}{\partial v}dv\\\\ \dfrac{\partial y}{\partial v}dv\end{pmatrix}}$$
よって
Ｐ'の座標：$${(dx_P,dy_P)}$$=$${\Big(\dfrac{\partial x}{\partial u}du,\dfrac{\partial y}{\partial u}du\Big)}$$
Ｑ'の座標：$${(dx_Q,dy_Q)}$$=$${\Big(\dfrac{\partial x}{\partial v}dv,\dfrac{\partial y}{\partial v}dv\Big)}$$
$${dxdy=}$$平行四辺形Ｏ'Ｐ'Ｑ'Ｒ' の面積
$${=\begin{vmatrix}dx_Pdy_Q-dx_Qdy_P\end{vmatrix}=\begin{vmatrix}\dfrac{\partial x}{\partial u}du\dfrac{\partial y}{\partial v}dv-\dfrac{\partial x}{\partial v}dv\dfrac{\partial y}{\partial u}du \end{vmatrix}}$$
$${=\begin{vmatrix}\dfrac{\partial x}{\partial u}\dfrac{\partial y}{\partial v}-\dfrac{\partial x}{\partial v}\dfrac{\partial y}{\partial u}\end{vmatrix}dudv}$$
ここで
$${\begin{vmatrix}\dfrac{\partial x}{\partial u}\dfrac{\partial y}{\partial v}-\dfrac{\partial x}{\partial v}\dfrac{\partial y}{\partial u}\end{vmatrix}=\begin{vmatrix}J\end{vmatrix}}$$$${\Big(=\dfrac{\partial(x,y)}{\partial(u,v)}}$$とも書く$${\Big)}$$
と定義すると
$${dxdy=\begin{vmatrix}J\end{vmatrix}dudv}$$である。
よって $${\displaystyle\int\displaystyle\int_Df(x,y)dxdy=\displaystyle\int_a^b\displaystyle\int_c^dg(u,v) \begin{vmatrix}J\end{vmatrix}dudv= …}$$