The integral notation $$\int_{\mathbb{R}^n}f\, \mathrm{d}^n x$$ represents the multiple integration of a function f over n-dimensional space. Specifically, it can be expressed as $$\int_\mathbb{R}\int_\mathbb{R}\ldots\int_\mathbb{R}f \,dx_1\,dx_2\ldots\,dx_n$$, indicating that each integration is performed with respect to a different variable. The variables $$dx_i$$ are distinct dummy variables, similar to the notation used in double integrals like $$\int \int f(x, y) dx dy$$.
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It means ##\int_\mathbb{R}\int_\mathbb{R}\ldots\int_\mathbb{R}f \,dx_1\,dx_2\ldots\,dx_n##Leo Liu said:Homework Statement:: .
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I saw it somewhere but I did't know exactly what it meant. Could someone explain it to me like I am 5? Does it mean we integrate with respect to x twice?
$$\int_{\mathbb{R}^n}f\, \mathrm{d}^n x$$
Thanks. Just need some clarification -- do x-n s represent the same parameter or different variables?fresh_42 said:It means ##\int_\mathbb{R}\int_\mathbb{R}\ldots\int_\mathbb{R}f \,dx_1\,dx_2\ldots\,dx_n##
Different variables. The integral is over a region in ##\mathbb R^n##. Each ##dx_i## is a different dummy variable, much the same as ##\int \int f(x, y) dx dy##.Leo Liu said:Thanks. Just need some clarification -- do x-n s represent the same parameter or different variables?