2 dimensional case

Submitted by Structure on Sat, 09/22/2007 - 20:01.

In this case we may write the name of the variable (x,y)
let $h(x,y)=g(a(x,y),b(x,y))$
Then

$\frac{\partial h}{\partial x}=\frac{\partial g}{\partial a}(a,b)\frac{\partial a}{\partial x}+\frac{\partial g}{\partial b}(a,b)\frac{\partial b}{\partial x}$

$\frac{\partial h}{\partial y}=\frac{\partial g}{\partial a}(a,b)\frac{\partial a}{\partial y}+\frac{\partial g}{\partial b}(a,b)\frac{\partial b}{\partial y}$

$\frac{\partial ^2h}{\partial x^2}= \frac{\partial ^2g}{\partial a^2}(a,b){\frac{\partial a}{\partial x}}{\frac{\partial a}{\partial x}}+ \frac{\partial ^2g}{\partial b\partial a}(a,b)\frac{\partial b}{\partial x}\frac{\partial a}{\partial x}+ \frac{\partial ^2g}{\partial a\partial b}(a,b)\frac{\partial a}{\partial x}\frac{\partial b}{\partial x}+ \frac{\partial ^2g}{\partial b^2}(a,b)\frac{\partial b}{\partial x}\frac{\partial b}{\partial x}+ \frac{\partial g}{\partial a}(a,b)\frac{\partial ^2 a}{\partial ^2x}+\frac{\partial g}{\partial b}(a,b)\frac{\partial ^2b}{\partial x^2}$

$\frac{\partial ^2h}{\partial x\partial y}= \frac{\partial ^2g}{\partial a^2}(a,b){\frac{\partial a}{\partial x}}{\frac{\partial a}{\partial y}}+ \frac{\partial ^2g}{\partial b\partial a}(a,b)\frac{\partial b}{\partial x}\frac{\partial a}{\partial y}+ \frac{\partial ^2g}{\partial a\partial b}(a,b)\frac{\partial a}{\partial x}\frac{\partial b}{\partial y}+ \frac{\partial ^2g}{\partial b^2}(a,b)\frac{\partial b}{\partial x}\frac{\partial b}{\partial y}+ \frac{\partial g}{\partial a}(a,b)\frac{\partial ^2 a}{\partial x\partial y}+\frac{\partial g}{\partial b}(a,b)\frac{\partial ^2b}{\partial x\partial y}$

$\frac{\partial ^2h}{\partial y^2}= \frac{\partial ^2g}{\partial a^2}(a,b){\frac{\partial a}{\partial y}}{\frac{\partial a}{\partial y}}+ \frac{\partial ^2g}{\partial b\partial a}(a,b)\frac{\partial b}{\partial y}\frac{\partial a}{\partial y}+ \frac{\partial ^2g}{\partial a\partial b}(a,b)\frac{\partial a}{\partial y}\frac{\partial b}{\partial y}+ \frac{\partial ^2g}{\partial b^2}(a,b)\frac{\partial b}{\partial y}\frac{\partial b}{\partial y}+ \frac{\partial g}{\partial a}(a,b)\frac{\partial ^2 a}{\partial ^2y}+\frac{\partial g}{\partial b}(a,b)\frac{\partial ^2b}{\partial y^2}$

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2 dimensional case