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Change of variable issue

  1. Feb 17, 2014 #1
    Hi there,

    I got across the integral
    $$\int_{\omega} \nabla y(x) \mathrm{d}x$$.
    It would be better to perform the integration over the domain $$\Omega$$, the two domains being related by a transformation $$Y:\omega \to \Omega$$.
    Using the change of variable rule I wrote
    $$\int_{\Omega} \nabla y(Y(x)) det\nabla Y \mathrm{d}x $$.
    Now before thinking how to write the gradient of a composite vectorial function I compared my computation to the paper I am trying to understand and noted that the result is written as
    $$\int_{\Omega} y_{Y} det\nabla Y \mathrm{d}x $$, where the notation $$y_{Y}$$ stands for the partial derivatives of y with respect to Y.
    Cannot make sense of it at all. The gradient of composite function should be
    $$\nabla (f \circ g) = (\nabla g)^{T} \cdot \nabla f$$, so could not figure out where the missing term in the correct computation (the paper one) is.

    Many thanks for your help as usual, the most appreciated.
  2. jcsd
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