Jacobian matrix generalization in coordinate transformation

In summary, there is a proof of the 'change of variable formula for integrals using Jacobians' which shows that the Jacobian matrix is compatible with higher dimensions.
  • #1
mertcan
345
6
hi, I always see that jacobian matrix is derived for just 2 dimension ( ıt means 2x2 jacobian matrix) in books while ensuring the coordinate transformation. After that kind of derivation, books say that you can use same principle for higher dimensions. But, I really wonder if there is a proof which ensure that jacobian matrix is compatible with higher dimensions? I am asking because there are proofs which ensure the jacobian matrix only in 2 dimensions not higher dimensions.. Thanks in advance... I am looking forward to your mathematical demonstrations...
 
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  • #2
Jacobian matrices have several useful properties.
For which one of those properties are you seeking a proof?
 
  • #3
andrewkirk said:
Jacobian matrices have several useful properties.
For which one of those properties are you seeking a proof?
While doing integral in terms of space-time coordinates, and if you want to change coordinates, we should use jacobian matrix and determinant rule. I know the proof of why we should use jacobian determinant rule if there are 2 coordinates, but I do not know the proof of how this jacobian determinant rule fits with the higher dimensions or high order coordinates ?
 
  • #4
I hope my question has become explicit after my last post...
 
  • #5
Is there someone who can answer my question ? :D I am really looking forward to your answers ...
 
  • #6
The proof you are seeking is that of the 'change of variable formula for integrals using Jacobians'. It is long and complex and I expect it will only appear in fairly advanced vector calculus texts.

This Stack Exchange Q&A gives an intuitive overview of the proof (in the first answer) and also contains a reference to a text in which the full proof can be found (in the last answer).
 

Related to Jacobian matrix generalization in coordinate transformation

1. What is a Jacobian matrix and why is it important in coordinate transformation?

A Jacobian matrix is a square matrix of partial derivatives that represents the rate of change of one coordinate system with respect to another. It is important in coordinate transformation because it allows us to understand how a change in one set of coordinates will affect the other set of coordinates.

2. How is the Jacobian matrix used in practical applications?

The Jacobian matrix is commonly used in fields such as physics, engineering, and computer graphics to solve problems involving coordinate transformations. It is also used in machine learning algorithms and optimization problems.

3. What is the relationship between the Jacobian matrix and the determinant?

The determinant of the Jacobian matrix represents the scaling factor of the coordinate transformation. It tells us how the volume changes when we switch from one coordinate system to another. A positive determinant indicates an expansion, while a negative determinant indicates a contraction.

4. Is the Jacobian matrix always invertible?

No, the Jacobian matrix is not always invertible. It is only invertible if the partial derivatives are all non-zero. In some cases, the Jacobian matrix may be singular, meaning it has a determinant of 0, and therefore cannot be inverted. This can happen when there is a degeneracy in the coordinate transformation.

5. Can the Jacobian matrix be extended to higher dimensions?

Yes, the concept of the Jacobian matrix can be extended to higher dimensions. In three-dimensional space, for example, the Jacobian matrix becomes a 3x3 matrix of partial derivatives. In general, the Jacobian matrix can be extended to any number of dimensions, but the calculation becomes more complex as the number of dimensions increases.

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