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").

**Physics Forums - The Fusion of Science and Community**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Change of variables in integrand

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

Loading...

Similar Threads - Change variables integrand | Date |
---|---|

B Some doubts about functions... (changing the independent variable from time to position) | May 6, 2017 |

I Change of variable and Jacobians | Feb 25, 2017 |

I Multi-dimensional Integral by Change of Variables | Feb 12, 2017 |

I Chain rule and change of variables again | Jan 11, 2017 |

**Physics Forums - The Fusion of Science and Community**