【微分積分】8-2-4 一般の変数変換｜要点まとめ

\(D\)を\(x=3u,y=4v\)で変数変換すると
\(\displaystyle E=\{(u,v)|u^2+v^2\leqq 1\}\)
ヤコビアンは
\(\displaystyle \frac{\partial(x,y)}{\partial(u,v)}=\begin{vmatrix}3 & 0 \\ 0 & 4\end{vmatrix}=12\)
すなわち
\(\displaystyle \iint_D(x^2+y^2)dxdy=\iint_E(9u^2+16v^2)・12dudv\)
\(E\)を極座標変換すると
\(\displaystyle F=\left\{(r,\theta)|0\leqq r\leqq 1,0\leqq \theta\leqq 2\pi\right\}\)
よって
\(\displaystyle \iint_D(x^2+y^2)dxdy\)
\(\displaystyle =12\iint_F(9r^2\cos^2\theta+16r^2\sin^2\theta)・rdrd\theta\)
\(\displaystyle =12\int_0^1r^3dr\int_0^{2\pi}(9+7\sin^2\theta)d\theta\)
\(\displaystyle =12・\frac{1}{4}\int_0^{2\pi}\frac{25-7\cos2\theta}{2}d\theta\)
\(=75\pi\)

\(D\)を\(s=y-2x,t=x+y\)で変数変換すると
\(\displaystyle E=\{(s,t)|0\leqq s\leqq 1,0\leqq t\leqq 1\}\)
ヤコビアンは
\(\displaystyle \frac{\partial(x,y)}{\partial(s,t)}=\begin{vmatrix}\displaystyle -\frac{1}{3} & \displaystyle \frac{1}{3} \\ \displaystyle \frac{1}{3} & \displaystyle \frac{2}{3}\end{vmatrix}=-\frac{1}{3}\)
よって
\(\displaystyle \iint_Dydxdy\)
\(\displaystyle =\iint_E\frac{s+2t}{3}・\left|-\frac{1}{3}\right|dsdt\)
\(\displaystyle =\frac{1}{9}\int_0^1\int_0^1(s+2t)dtds\)
\(\displaystyle =\frac{1}{9}\int_0^1(s+1)ds\)
\(\displaystyle =\frac{1}{6}\)