Jacobian and chain rule
Let , be differentiable functions of , and , be differentiable functions of , . Then the connection
between the Jacobian determinants is in .
Proof. Starting from the right hand side of (1), where one can multiply the determinants (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 rule (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 variables.
|Title||Jacobian and chain rule|
|Date of creation||2013-03-22 18:59:45|
|Last modified on||2013-03-22 18:59:45|
|Last modified by||pahio (2872)|