Doing the chain rule

*To*: mathgroup at smc.vnet.net*Subject*: [mg128725] Doing the chain rule*From*: tsg.moore at gmail.com*Date*: Wed, 21 Nov 2012 19:47:54 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*Delivered-to*: l-mathgroup@wolfram.com*Delivered-to*: mathgroup-newout@smc.vnet.net*Delivered-to*: mathgroup-newsend@smc.vnet.net

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?