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

  • #1
George Keeling
Gold Member
73
10

Main Question or Discussion Point

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

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

Could the integral also be written as follows?
longintegral.png
 

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Answers and Replies

  • #2
1,013
65
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.)
 
  • #3
George Keeling
Gold Member
73
10
I did indeed mean dx dy dz dt. Thank you, I liked your answer!
 
  • #5
George Keeling
Gold Member
73
10
That's enough thanks!
 

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