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.

Could the integral also be written as follows?

The curly L is a Lagrange density.

*S*is an action, Φ is a vector potential.Could the integral also be written as follows?

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slider142 replied on physicsforums:

ReplyDelete"Yes, assuming you mean dx dy dz dt. One reason for using d4x 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.)"

Demystifier added

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

A very long thread on the subject.