- #1
Barnak
- 63
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I'm having an interpretation problem with the notation used in physics, under the integration sign.
What is the proper interpretation of the domain of integration symbol, on the integration sign ?
To be more precise, consider a function [itex]F(x)[/itex] of one or several variables. Its integral on a domain [itex]D[/itex] is simply
[itex]I = \int_D F(x) \, dx[/itex].
Now, we procede to a change of variables : [itex]x \rightarrow x' = h(x)[/itex], so the function is now a new function of the new variables : [itex]F(x) \rightarrow G(x')[/itex]. Of course, the integral above gives the same number, but the limits of integration have to change to adapt to the new variables. We can write :
[itex]I \equiv \int_{D'} G(x') \, dx'[/itex].
My question is this : is the symbol [itex]D[/itex] on the integration sign actually an invariant, so [itex]D' \equiv D[/itex] ? Or is it a representation of the values that the variables are taking under the integral (so [itex]D' \ne D[/itex]) ?
In other words : do the domain of integration [itex]D[/itex] change with the coordinates transformation ?
What is the proper interpretation of the domain of integration symbol, on the integration sign ?
To be more precise, consider a function [itex]F(x)[/itex] of one or several variables. Its integral on a domain [itex]D[/itex] is simply
[itex]I = \int_D F(x) \, dx[/itex].
Now, we procede to a change of variables : [itex]x \rightarrow x' = h(x)[/itex], so the function is now a new function of the new variables : [itex]F(x) \rightarrow G(x')[/itex]. Of course, the integral above gives the same number, but the limits of integration have to change to adapt to the new variables. We can write :
[itex]I \equiv \int_{D'} G(x') \, dx'[/itex].
My question is this : is the symbol [itex]D[/itex] on the integration sign actually an invariant, so [itex]D' \equiv D[/itex] ? Or is it a representation of the values that the variables are taking under the integral (so [itex]D' \ne D[/itex]) ?
In other words : do the domain of integration [itex]D[/itex] change with the coordinates transformation ?