【微分積分】7-4-1 合成関数の微分｜要点まとめ

\(g_x(x,y)\)
\(\displaystyle =f'(x^2-y^2)・\frac{\partial}{\partial x}(x^2-y^2)\)
\(\displaystyle =2xf'(x^2-y^2)\)
\(g_y(x,y)\)
\(\displaystyle =f'(x^2-y^2)・\frac{\partial}{\partial y}(x^2-y^2)\)
\(\displaystyle =-2yf'(x^2-y^2)\)
よって、
\(yg_x(x,y)+xg_y(x,y)\)
\(=y・2xf'(x^2-y^2)+x・(-2y)f'(x^2-y^2)\)
\(=0\)

\(h_x(x,y)\)
\(\displaystyle =f'\left(\frac{y}{x}\right)・\frac{\partial}{\partial x}\frac{y}{x}\)
\(\displaystyle =-\frac{y}{x^2}f'\left(\frac{y}{x}\right)\)
\(h_y(x,y)\)
\(\displaystyle =f'\left(\frac{y}{x}\right)・\frac{\partial}{\partial y}\frac{y}{x}\)
\(\displaystyle =\frac{1}{x}f'\left(\frac{y}{x}\right)\)
よって、
\(xh_x(x,y)+yh_y(x,y)\)
\(\displaystyle =x・\frac{-y}{x^2}f'\left(\frac{y}{x}\right)+y・\frac{1}{x}f'\left(\frac{y}{x}\right)\)
\(=0\)

\(x'=2t,y'=3t^2\)より
\(g'(t)\)
\(\displaystyle =f_x(t^2+1,t^3)\frac{dx}{dt}+f_y(t^2+1,t^3)\frac{dy}{dt}\)
\(\displaystyle =2tf_x(t^2+1,t^3)+3t^2f_y(t^2+1,t^3)\)
よって、
\(g''(t)\)
\(=2f_x(t^2+1,t^3)\)
\(\ +2t\{f_{xx}(t^2+1,t^3)・2t+f_{xy}(t^2+1,t^3)・3t^2\}\)
\(\ +6tf_y(t^2+1,t^3)\)
\(\ +3t^2\{f_{yx}(t^2+1,t^3)・2t+f_{yy}(t^2+1,t^3)・3t^2\}\) \(=4t^2f_{xx}(t^2+1,t^3)+12t^3f_{xy}(t^2+1,t^3)\)
\(\ +9t^4f_{yy}(t^2+1,t^3)+2f_x(t^2+1,t^3)\)
\(\ +6tf_y(t^2+1,t^3)\)

\(E(t)\)を微分すると
\(E'(t)\)
\(\displaystyle =\frac{1}{2}m\{2x'(t)x''(t)+2y'(t)y''(t)\}\)
\(\ +V_x(x(t),y(t))x'(t)+V_y(x(t),y(t))y'(t)\)
\(\displaystyle =mx''(t)x'(t)+my''(t)y'(t)\)
\(\ +V_x(x(t),y(t))x'(t)+V_y(x(t),y(t))y'(t)\)
\(\displaystyle =-V_x(x(t),y(t))x'(t)-V_y(x(t),y(t))y'(t)\)
\(\ +V_x(x(t),y(t))x'(t)+V_y(x(t),y(t))y'(t)\)
\(\displaystyle =0\)
よって、
\(E(t)\)は\(t\)によらない定数関数である。

\(g_u(u,v)=2uf_x(u^2+v^2,uv)+vf_y(u^2+v^2,uv)\)
\(g_v(u,v)=2vf_x(u^2+v^2,uv)+uf_y(u^2+v^2,uv)\)
よって、
\(vg_u(u,v)+ug_v(u,v)\)
\(=v\{2uf_x(u^2+v^2,uv)+vf_y(u^2+v^2,uv)\}\)
\(\ +u\{2vf_x(u^2+v^2,uv)+uf_y(u^2+v^2,uv)\}\)
\(=4uvf_x(u^2+v^2,uv)+(u^2+v^2)f_y(u^2+v^2,uv)\)

\(v_r(r,\theta)\)
\(=u_x(r\cos\theta,r\sin\theta)\cos\theta\)
\(\ \ \ +u_y(r\cos\theta,r\sin\theta)\sin\theta\)
\(v_\theta(r,\theta)\)
\(=u_x(r\cos\theta,r\sin\theta)(-r\sin\theta)\)
\(\ \ \ +u_y(r\cos\theta,r\sin\theta)r\cos\theta\)
よって、
\(\displaystyle v_r^2+\frac{1}{r^2}v_\theta^2\)
\(=(u_x\cos\theta+u_y\sin\theta)^2\)
\(\displaystyle \ +\frac{1}{r^2}(-u_xr\sin\theta+u_yr\cos\theta)^2\)
\(=u_x^2(\cos^2\theta+\sin^2\theta)+u_y^2(\sin^2\theta+\cos^2\theta)\)
\(=u_x^2+u_y^2\)

\(g_s=f_xe^s\cos t+f_ye^s\sin t\)
\(u_t=f_x・(-e^s\sin t)+f_ye^s\cos t\)
\(g_{ss}\)
\(=(f_{xx}e^s\cos t+f_{xy}e^s\sin t)e^s\cos t+f_xe^s\cos t\)
\(\ +(f_{yx}e^s\cos t+f_{yy}e^s\sin t)e^s\sin t+f_ye^s\sin t\)
\(=e^{2s}(f_{xx}\cos^2t+f_{xy}\sin2t+f_{yy}\sin^2t)\)
\(\ +e^s(f_x\cos t+f_y\sin t)\)
\(g_{tt}\)
\(=\{f_{xx}・(-e^s\sin t)+f_{xy}e^s\cos t\}(-e^s\sin t)\)
\(\ +f_x・(-e^s\cos t)\)
\(\ +\{f_{yx}・(-e^s\sin t)+f_{yy}e^s\cos t\}e^s\cos t\)
\(\ +f_y・(-e^s\sin t)\)
\(=e^{2s}(f_{xx}\sin^2t-f_{xy}\sin2t+f_{yy}\cos^2t)\)
\(\ -e^s(f_x\cos t+f_y\sin t)\)
\(f(x,y)\)は調和関数なので、
\(\Delta g(s,t)\)
\(=g_{ss}+g_{tt}\)
\(=e^{2s}(f_{xx}+f{yy})\)
\(=0\)
よって、\(g(s,t)\)は調和関数である。

\(v_r=u_x\cos\theta+u_y\sin\theta\)
\(v_\theta=u_x・(-r\sin\theta)+u_yr\cos\theta\)
\(v_{rr}\)
\(=(u_{xx}\cos\theta+u_{xy}\sin\theta)\cos\theta\)
\(\ \ \ +(u_{yx}\cos\theta+u_{yy}\sin\theta)\sin\theta\)
\(=u_{xx}\cos^2\theta+u_{yy}\sin^2\theta+u_{xy}\sin2\theta\)
\(v_{\theta\theta}\)
\(=\{u_{xx}・(-r\sin\theta)+u_{xy}r\cos\theta\}・(-r\sin\theta)\)
\(\ \ \ +u_x・(-r\cos\theta)\)
\(\ \ \ +\{u_{yx}・(-r\sin\theta)+u_{yy}r\cos\theta\}r\cos\theta\)
\(\ \ \ +u_y・(-r\sin\theta)\)
\(=r^2(u_{xx}\sin^2\theta+u_{yy}\cos^2\theta-u_{xy}\sin2\theta)\)
\(\ \ \ -r(u_x\cos\theta+u_y\sin\theta)\)
よって、
\(\displaystyle v_{rr}+\frac{1}{r}v_r+\frac{1}{r^2}v_{\theta\theta}\)
\(=(u_{xx}\cos^2\theta+u_{yy}\sin^2\theta+u_{xy}\sin2\theta)\)
\(\displaystyle \ +\frac{1}{r}(u_x\cos\theta+u_y\sin\theta)\)
\(\displaystyle \ +\frac{1}{r^2}\{r^2(u_{xx}\sin^2\theta+u_{yy}\cos^2\theta-u_{xy}\sin2\theta)\)
\(\displaystyle \ \ \ -r(u_x\cos\theta+u_y\sin\theta)\}\)
\(=u_{xx}(\cos^2\theta+\sin^2\theta)+u_{yy}(\cos^2\theta+\sin^2\theta)\)
\(=u_{xx}+u_{yy}\)