Help with integration conventions in "Spacetime and Geometry" by Sean Carroll

[George Keeling]

I am reading Spacetime and Geometry by Sean Carroll. In section 1.10 on classical field theory, he uses this formula (1.132)

The curly L is a Lagrange density. S is an action, Φ is a vector potential.

Could the integral also be written as follows?

 

[slider142]

Yes, assuming you mean $dx\, dy\, dz\, dt$. One reason for using $d^4x$ instead is to save space, and to not emphasize any particular order for the iterated integrals. (There is also a little matter of the fact that the integral over a region is only equivalent to iterated integration over a series of intervals when the region and function being integrated satisfy certain properties, such as Fubini's theorem, but this is not really a big deal most of the time.)

 

[George Keeling]

I did indeed mean dx dy dz dt. Thank you, I liked your answer!

 

[Demystifier]

See also https://www.physicsforums.com/threads/dx-before-the-f-x-in-integrals.950465/

 

[George Keeling]

That's enough thanks!