1.次の関数の\(f_t(x,y)\)を求めなさい。ただし、\(f(x,y)\)は\(C^1\)級とする。
(1)\(f(x,y)=x^2+2y,\ x=2t,\ y=t^3\)
\(\displaystyle f_t(x,y)=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}\)
\(=2x・2+2・3t^2\)
\(=8t+6t^2\)
(2)\(f(x,y)=x^2+y^2,\ x=\cos t,\ y=\sin t\)
\(\displaystyle f_t(x,y)=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}\)
\(=2x(-\sin t)+2y\cos t\)
\(=-2\sin t\cos t+2\sin t\cos t\)
\(=0\)
(3)\(f(x,y)=x^2+xy+2y^2,\ x=\cos t,\ y=\sin t\)
\(\displaystyle f_t(x,y)=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}\)
\(=(2x+y)(-\sin t)+(x+4y)\cos t\)
\(=(2\cos t+\sin t)(-\sin t)+(\cos t+4\sin t)\cos t\)
\(=\cos^2t-\sin^2t+2\sin t\cos t\)
\(=\cos2t+\sin2t\)
(4)\(f(x,y)=x^3y^2,\ x=t^2,\ y=t^3\)
\(\displaystyle f_t(x,y)=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}\)
\(=3x^2y^2・2t+2x^3y・3t^2\)
\(=3t^4t^6・2t+2t^6t^3・3t^2\)
\(=6t^{11}+6t^{11}\)
\(=12t^{11}\)
(5)\(\displaystyle f(x,y)=\log(x^2+y^2),\ x=t+\frac{1}{t},\ y=t(t-1)\)
\(\displaystyle f_t(x,y)=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}\)
\(\displaystyle =\frac{2x}{x^2+y^2}\left(1-\frac{1}{t^2}\right)+\frac{2y}{x^2+y^2}(2t-1)\)
(6)\(f(x,y),\ x=t^2,\ y=e^t\)
\(\displaystyle f_t(x,y)=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}\)
\(\displaystyle =2tf_x(t^2,e^t)+e^tf_y(t^2,e^t)\)
(7)\(f(x,y),\ x=2t,\ y=4t^2\)
\(\displaystyle f_t(x,y)=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}\)
\(\displaystyle =2f_x(2t,4t^2)+8tf_y(2t,4t^2)\)
(8)\(f(x,y)=x^2-2y^2,\ x=\cos t,\ y=\sin t\)
\(\displaystyle f_t(x,y)=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}\)
\(=2x(-\sin t)-4y\cos t\)
\(=-2\sin t\cos t-4\sin t\cos t\)
\(=-6\sin t\cos t\)
\(=-3\sin2t\)
2.次の関数の\(f_u(x,y),f_v(x,y)\)を求めなさい。ただし、\(f(x,y)\)は\(C^1\)級とする。
(1)\(f(x,y)=x^2+y^2,\ x=u-2v,\ y=2u+v\)
\(\displaystyle f_u(x,y)=\frac{\partial f}{\partial x}\frac{dx}{du}+\frac{\partial f}{\partial y}\frac{dy}{du}\)
\(=2x+2y・2\)
\(=2(u-2v)+4(2u+v)\)
\(=10u\)
\(\displaystyle f_v(x,y)=\frac{\partial f}{\partial x}\frac{dx}{dv}+\frac{\partial f}{\partial y}\frac{dy}{dv}\)
\(=2x・(-2)+2y\)
\(=-4(u-2v)+2(2u+v)\)
\(=10v\)
(2)\(f(x,y)=x^2+xy+2y^2,\ x=u+v,\ y=uv\)
\(\displaystyle f_u(x,y)=\frac{\partial f}{\partial x}\frac{dx}{du}+\frac{\partial f}{\partial y}\frac{dy}{du}\)
\(=2x+y+(x+4y)v\)
\(=2(u+v)+uv+(u+v+4uv)v\)
\(=2u+2v+2uv+v^2+4uv^2\)
\(\displaystyle f_v(x,y)=\frac{\partial f}{\partial x}\frac{dx}{dv}+\frac{\partial f}{\partial y}\frac{dy}{dv}\)
\(=2x+y+(x+4y)u\)
\(=2(u+v)+uv+(u+v+4uv)u\)
\(=2u+2v+2uv+u^2+4u^2v\)
(3)\(f(x,y)=x^2y^2,\ x=uv,\ y=v^2\)
\(\displaystyle f_u(x,y)=\frac{\partial f}{\partial x}\frac{dx}{du}+\frac{\partial f}{\partial y}\frac{dy}{du}\)
\(=2xy^2・v\)
\(=2uv・v^4・v\)
\(=2uv^6\)
\(\displaystyle f_v(x,y)=\frac{\partial f}{\partial x}\frac{dx}{dv}+\frac{\partial f}{\partial y}\frac{dy}{dv}\)
\(=2xy^2・u+2x^2y・2v\)
\(=2uv(v^2)^2・u+2(uv)^2v^2・2v\)
\(=6u^2v^5\)
(4)\(\displaystyle f(x,y)=\tan^{-1}\frac{y}{x},\ x=u^3-3uv^2,\ y=3u^2v-v^3\)
\(\displaystyle f(x,y)=\tan^{-1}\frac{y}{x}=3\tan^{-1}\frac{v}{u}\)
\(\displaystyle f_u(x,y)=3・\frac{1}{1+(\frac{v}{u})^2}\left(-\frac{v}{u^2}\right)\)
\(\displaystyle =-\frac{3v}{u^2+v^2}\)
\(\displaystyle f_v(x,y)=3・\frac{1}{1+(\frac{v}{u})^2}\left(\frac{1}{u}\right)\)
\(\displaystyle =-\frac{3u}{u^2+v^2}\)
(5)\(\displaystyle f(x,y)=\log\frac{y}{x},\)
\(\ x=(u-1)^2+v^2,\ y=(u+1)^2+v^2\)
\(\displaystyle f_u(x,y)=\frac{\partial f}{\partial x}\frac{dx}{du}+\frac{\partial f}{\partial y}\frac{dy}{du}\)
\(\displaystyle =\frac{-\frac{y}{x^2}}{\frac{y}{x}}・2(u-1)+\frac{\frac{1}{x}}{\frac{y}{x}}・2(u+1)\)
\(\displaystyle =\frac{-2(u-1)}{x}+\frac{2(u+1)}{y}\)
\(\displaystyle =\frac{-2(u-1)}{(u-1)^2+v^2}+\frac{2(u+1)}{(u+1)^2+v^2}\)
\(\displaystyle f_v(x,y)=\frac{\partial f}{\partial x}\frac{dx}{dv}+\frac{\partial f}{\partial y}\frac{dy}{dv}\)
\(\displaystyle =\frac{-\frac{y}{x^2}}{\frac{y}{x}}・2v+\frac{\frac{1}{x}}{\frac{y}{x}}・2v\)
\(\displaystyle =\frac{-2v}{x}+\frac{2v}{y}\)
\(\displaystyle =\frac{-2v}{(u-1)^2+v^2}+\frac{2v}{(u+1)^2+v^2}\)
(6)\(f(x,y)=\sqrt{x^2+y^2},\ x=u\cos v,\ y=u\sin v\)
\(f(x,y)=\sqrt{x^2+y^2}=|u|\)
\(f_u(x,y)=1\)
\(f_v(x,y)=0\)
(7)\(f(x,y),\ x=u\cos v,\ y=u\sin v\)
\(\displaystyle f_u(x,y)=\frac{\partial f}{\partial x}\frac{dx}{du}+\frac{\partial f}{\partial y}\frac{dy}{du}\)
\(=f_x(u\cos v,u\sin v)\cos v\)
\(\ +f_y(u\cos v,u\sin v)\sin v\)
\(\displaystyle f_v(x,y)=\frac{\partial f}{\partial x}\frac{dx}{dv}+\frac{\partial f}{\partial y}\frac{dy}{dv}\)
\(=-uf_x(u\cos v,u\sin v)\sin v\)
\(\ +uf_y(u\cos v,u\sin v)\cos v\)
