107 學年度台綜大微積分 A 第 7 題

題目

7. (10 Points) Let

f:\mathbb{R}^2\to\mathbb{R}

be a function. Assume that all second partial derivatives of

f

exist and continuous and

f_{xx}+f_{yy}=0,\quad \text{for all }(x,y)\in\mathbb{R}^2.

Define $g:\mathbb{R}^2\to\mathbb{R}$ by $g(u,v)=f(u^2-v^2,2uv)$ . Find $g_{uu}+g_{vv}$ .

令 $X=u^2-v^2$ 、 $Y=2uv$ 。兩次套用鏈鎖律可得

g_{uu}+g_{vv} =4(u^2+v^2)(f_{xx}+f_{yy})\big|_{(X,Y)}.

題設 $f_{xx}+f_{yy}=0$ ，因此 $\boxed{g_{uu}+g_{vv}=0}$ 。