How to Understand the Change of Variable in a Vectorial Function Integral?

[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.

 

[jvicens]

Hello!

Thank you for sharing your integral and your thoughts on it. It seems like you have a good understanding of the change of variable rule and the gradient of a composite function. However, in the paper you are trying to understand, the notation $$y_{Y}$$ is not referring to the partial derivatives of y with respect to Y. It is actually a shorthand notation for the function y composed with the transformation Y. In other words, $$y_{Y} = y \circ Y$$.

With this understanding, the integral in the paper can be rewritten as $$\int_{\Omega} (y \circ Y) det\nabla Y \mathrm{d}x$$, which is equivalent to your previous computation. The missing term is simply the function y being composed with the transformation Y. I hope this clarifies things for you. Keep up the good work in your studies!