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)
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- Delivered-to: l-mathgroup@wolfram.com
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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?