Understanding Tensor Integration Notation for 4-Dimensional Space

[superg33k]

Hi,

I have been going round in circles trying to figure it out so I figured it would be quickest and easiest to just ask:

What is the definition of: $$\int f(x) d^4 x$$

I first thought it was:

$$=\int \int \int \int f(x) dx^1 dx^2 dx^3 dx^4$$

but now I am starting to think it is:

$$= \int f(x) dx^1 +\int f(x) dx^2 +\int f(x) dx^3 +\int f(x) dx^4$$

Thanks for any help.

 

[dextercioby]

Of course it's the first one. It's an integral over a 4D space/manifold, your second equality makes no sense.

 

[andrien]

In general a factor of √g is absent in the former expression ,the second is just weird.