Jacobian and chain rule

Let u, v be differentiable functions of x, y and x, y be differentiable functions of s, t.  Then the connection

(u,v)(s,t)=(u,v)(x,y)(x,y)(s,t) (1)

between the Jacobian determinants is in .

Proof.  Starting from the right hand side of (1), where one can multiply the determinantsMathworldPlanetmath (http://planetmath.org/Determinant2) similarly as the corresponding matrices (http://planetmath.org/MatrixMultiplication), we have


Here, the last stage has been written according to the general chain ruleMathworldPlanetmath (http://planetmath.org/ChainRuleSeveralVariables).  But thus we have arrived at the left hand side of the equation (1), which hereby has been proved.

Remark.  The rule (1) is only a visualisation of the more general one concerning the case of functions of n variables.

Title Jacobian and chain rule
Canonical name JacobianAndChainRule
Date of creation 2013-03-22 18:59:45
Last modified on 2013-03-22 18:59:45
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 4
Author pahio (2872)
Entry type Example
Classification msc 15-00
Classification msc 26B05
Classification msc 26B10