- #1
Nick R
- 69
- 0
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").
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").
