【微分積分】7-3-1 高階偏導関数の定義｜問題集

1階偏導関数を求めると
\(f_x(x,y)=ye^{xy}\)
\(f_y(x,y)=xe^{xy}\)
よって、2階偏導関数は
\(\displaystyle f_{xx}(x,y)=y^2e^{xy}\)
\(\displaystyle f_{xy}(x,y)=f_{yx}(x,y)=(1+xy)e^{xy}\)
\(\displaystyle f_{yy}(x,y)=x^2e^{xy}\)

1階偏導関数を求めると
\(f_x(x,y)=2x\cos(x^2+y^2)\)
\(f_y(x,y)=2y\cos(x^2+y^2)\)
よって、2階偏導関数は
\(\displaystyle f_{xx}(x,y)=2\cos(x^2+y^2)-4x^2\sin(x^2+y^2)\)
\(\displaystyle f_{xy}(x,y)=f_{yx}(x,y)=-4xy\sin(x^2+y^2)\)
\(\displaystyle f_{yy}(x,y)=2\cos(x^2+y^2)-4y^2\sin(x^2+y^2)\)

1階偏導関数を求めると
\(f_x(x,y)=e^x(x\cos y-y\sin y)+e^x\cos y\)
\(=e^x\{(x+1)\cos y-y\sin y\}\)
\(f_y(x,y)=e^x(-x\sin y-\sin y-y\cos y)\)
\(=-e^x\{y\cos y+(x+1)\sin y\}\)
2階偏導関数を求めると
\(f_{xx}(x,y)=e^x\{(x+1)\cos y-y\sin y\}+e^x\cos y\)
\(=e^x\{(x+2)\cos y-y\sin y\}\)
\(f_{yy}(x,y)=-e^x\{\cos y-y\sin y+(x+1)\cos y\}\)
\(=-e^x\{(x+2)\cos y-y\sin y\}\)
よって、ラプラシアンは
\(\Delta=f_{xx}(x,y)+f_{yy}(x,y)\)
\(\displaystyle =e^x\{(x+2)\cos y-y\sin y-e^x\{(x+2)\cos y-y\sin y\}\)
\(=0\)