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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.

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# Change of variable issue

Can you offer guidance or do you also need help?

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