演習問題

${\displaystyle z=f\left(x,y\right)}$, ${\displaystyle x=u\cos \alpha -v\sin \alpha }$, ${\displaystyle y=u\sin \alpha +v\cos \alpha }$ならば

${\displaystyle {\left(\frac{\partial z}{\partial x}\right)}^2+{\left(\frac{\partial z}{\partial y}\right)}^2}$ ${\displaystyle ={\left(\frac{\partial z}{\partial u}\right)}^2+{\left(\frac{\partial z}{\partial v}\right)}^2}$

となることを示せ．

次の問に答えよ．

(1)点${\displaystyle \text{P}\left(x,y\right)}$ を原点を中心に${\displaystyle 60°}$ 回転させ点${\displaystyle \text{Q}\left(u,v\right)}$ に移す1次変換を表す行列 ${\displaystyle A}$ を求めよ． 三角関数は計算すること．

(2) ${\displaystyle \text{Q}\left(u,v\right)}$ を ${\displaystyle y}$ 軸に関して対称な点 ${\displaystyle \text{R}(r,s)に}$ 移す.1次変換を表す行列 ${\displaystyle B}$ を求めよ．

(3)点${\displaystyle \text{P}\left(x,y\right)}$ を点${\displaystyle \text{R}\left(r,s\right)}$ に移す1次変換を表す行列${\displaystyle C}$ を求めよ．