Change of variables in integrand

1. Aug 3, 2010

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").