Change of variables in integrand

[Nick R]

I have been trying to understand and articulate why I can't do the following. Please confirm or point out misunderstanding.

There is an integral in the "hatted" system,

$$\int_{R}\bar{f}(\bar{x}^{1},...,\bar{x}^{n})d\bar{x}^{1}...d\bar{x}^{n}$$

I want to express this as an integral in the unhatted system. To this end, note that the following relationship is true:

$$d\bar{x}^{1}=\frac{\partial\bar{x}^{1}}{\partial x^{h}}dx^{h}$$
(einstein notation)

So the following is true:

$$\int_{R}\bar{f}(\bar{x}^{1},...,\bar{x}^{n})\frac{\partial\bar{x}^{1}}{\partial x^{h_{1}}}...\frac{\partial\bar{x}^{n}}{\partial x^{h_{n}}}dx^{h_{1}}...dx^{h_{n}}=\int_{R}\bar{f}(\bar{x}^{1},...,\bar{x}^{n})d\bar{x}^{1}...d\bar{x}^{n}$$

But the LHS is not what it seems to be (it isn't an integral in unhatted coordinates...) - using a different approach the "correct" expression is

$$\int_{R}\bar{f}(\bar{x}^{1},...,\bar{x}^{n})\left|\frac{\partial(\bar{x}^{1},...,\bar{x}^{n})}{\partial(x^{h_{1}},...,x^{h_{n}})}\right|dx^{h_{1}}...dx^{h_{n}}=\int_{R}\bar{f}(\bar{x}^{1},...,\bar{x}^{n})d\bar{x}^{1}...d\bar{x}^{n}$$

where the Jacobian is a "scale factor" to convert unhatted infintesimal volume element to hatted infintesimal volume element, obtained form the geometric interpretation of the determinate and $$d\bar{x}^{1}=\frac{\partial\bar{x}^{1}}{\partial x^{h}}dx^{h}$$

The reason the LHS of the "first attempt" is not what it seems to be is because

A change of variables in the integrand involves changing the arrangement, size and number of volume elements. So for

$$\int_{R}\bar{f}(\bar{x}^{1},...,\bar{x}^{n})\frac{\partial\bar{x}^{1}}{\partial x^{h_{1}}}...\frac{\partial\bar{x}^{n}}{\partial x^{h_{n}}}dx^{h_{1}}...dx^{h_{n}}=\int_{R}\bar{f}(\bar{x}^{1},...,\bar{x}^{n})d\bar{x}^{1}...d\bar{x}^{n}$$

to be true, $$dx^{h_{1}},...,$$ and $$dx^{h_{n}}$$ have to vary spatially so that they correspond to $$d\bar{x}^{1}, ...,$$ and $$d\bar{x}^{n}$$

Otherwise, what will effectively be happening is that volume elements will overlap (or "underlap").

 

[fresh_42]

In case you use differential forms, then you have to use the correct product, too, which is the Graßmann product. It will automatically give you the correct scaling factor, namely the functional determinant which you called 'Jacobian' in the above. Here is an example:
https://www.physicsforums.com/threa...an-and-polar-coordinates.980855/#post-6266564