# Change of variable issue

1. Feb 17, 2014

### muzialis

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.