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I have a transformation $$(x'_1,x'_2)=(f(x_1,x_2), g(x_1,x_2))$$ and I wish to find $$\partial x'_1\over \partial x'_2$$ how might I evaluate this?

If it is difficult to find a general expression for this, what if we suppose $f,g$ are simply linear combinations of $x_1,x_2$ so something like $ax_1+bx_2$ where $a,b$ are constants?

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You need to specify which variables you're considering as independent, i.e. which variables other than $x_2'$ you're keeping fixed. Are you looking for $$ \left.\frac{\partial x_1'}{\partial x_2'}\right|_{x_1} $$ or $$ \left.\frac{\partial x_1'}{\partial x_2'}\right|_{x_2}\;? $$ –  joriki Nov 25 '12 at 11:02
    
Thank you, @joriki . For example, if my variables are $(x',y')=(x+a, x+ay+b)$ , then what would $$\partial x'\over \partial y'$$ be? –  Fred Nov 25 '12 at 14:26
    
@Fred: you have not exactly answered joriki's question. –  Willie Wong Jan 8 '13 at 15:33
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