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

In summary, the conversation discusses the use of the change of variable rule for integrals and the notation for composite functions. The notation $$y_{Y}$$ in the paper is not referring to partial derivatives, but rather a shorthand for the function y composed with the transformation Y. This notation does not affect the overall result of the integral.
  • #1
muzialis
166
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.
 
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  • #2




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!
 

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

1. How does a change of variable affect the outcome of a scientific experiment?

A change of variable can significantly affect the outcome of a scientific experiment by altering the values and relationships between the variables being studied. It can either enhance the accuracy and precision of the results or introduce errors and bias.

2. Can a change of variable be made during an experiment or must it be decided beforehand?

A change of variable can be made during an experiment, but it is important to carefully plan and document any changes. This ensures that the results can be accurately interpreted and replicated by other scientists.

3. What are some common reasons for making a change of variable in a scientific study?

A change of variable may be necessary to correct for confounding factors, improve the sensitivity of the experiment, or better understand the relationships between variables. It may also be necessary due to limitations in equipment or resources.

4. How does a change of variable impact the statistical analysis of data?

A change of variable can affect the statistical analysis of data by altering the assumptions and methods used. It is important to carefully consider the implications of any changes on the statistical analysis to ensure accurate and meaningful results.

5. Are there any ethical considerations when making a change of variable in a scientific study?

Yes, there can be ethical considerations when making a change of variable in a scientific study. It is important to ensure that any changes do not compromise the safety or well-being of human or animal participants, and that they are conducted in accordance with ethical guidelines and regulations.

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