I have been trying to understand and articulate why I can't do the following. Please confirm or point out misunderstanding.(adsbygoogle = window.adsbygoogle || []).push({});

There is an integral in the "hatted" system,

[tex]\int_{R}\bar{f}(\bar{x}^{1},...,\bar{x}^{n})d\bar{x}^{1}...d\bar{x}^{n}[/tex]

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

[tex]d\bar{x}^{1}=\frac{\partial\bar{x}^{1}}{\partial x^{h}}dx^{h}[/tex]

(einstein notation)

So the following is true:

[tex]\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}[/tex]

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

[tex]\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}[/tex]

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 [tex]d\bar{x}^{1}=\frac{\partial\bar{x}^{1}}{\partial x^{h}}dx^{h}[/tex]

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

[tex]\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}[/tex]

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

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

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# Change of variables in integrand

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