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

  • #1
George Keeling
Gold Member
128
24
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
 

Attachments

  • shortintegral.png
    shortintegral.png
    1.7 KB · Views: 704
  • longintegral.png
    longintegral.png
    2.1 KB · Views: 389

Answers and Replies

  • #2
1,013
70
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.)
 
  • Like
Likes George Keeling
  • #3
George Keeling
Gold Member
128
24
I did indeed mean dx dy dz dt. Thank you, I liked your answer!
 
  • #5
George Keeling
Gold Member
128
24
That's enough thanks!
 

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

  • Last Post
Replies
4
Views
1K
Replies
3
Views
814
Replies
5
Views
649
  • Last Post
Replies
9
Views
2K
Replies
10
Views
1K
Replies
21
Views
2K
  • Last Post
Replies
16
Views
7K
Replies
10
Views
2K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
5
Views
1K
Top