MathGroup Archive 2012

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Doing the chain rule

Suppose I have a function u(x, y). I have a transformation from (X, Y) to (x, y) via u(x,y) = U(X,Y) and 

X = f(x,y) 
Y = g(x,y)

Using Mathematica, how do I compute the derivatives u_x, u_y, u_{xx}, u_{yy}, etc. in terms of U?

For example, 
u_x = f_x U_X + g_y U_Y

For example, if 
X = y - x
Y = y^2 - x^2

then u_x = -U_X + 2x U_Y. 

The problem here is that without knowing the inverses of f(x,y) and g(x,y), I don't know how to get Mathematica to work out the chain rules. For example, can you show me how to compute u_{xx} in mathematica using the above example? 

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