Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Expanding field derivative

  1. Jul 28, 2015 #1
    I just came across this in a textbook: ## (\partial_{\mu}\phi)^2 = (\partial_{\mu}\phi)(\partial^{\mu}\phi) ##

    Can someone explain why this makes sense? Thanks.
  2. jcsd
  3. Jul 28, 2015 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    The right-hand side is a short-hand way of writing
    $$\sum_{\mu=0}^d (\partial_{\mu}\phi)(\partial^{\mu}\phi).$$
    More generally, this shorthand is called the Einstein summation convention, commonly used whenever vectors and tensors appear, where a sum on a repeated index is assumed from the context of the expression. The left-hand side is a further shorthand, where if the quantity is supposed to be a scalar from the context, then it is assumed that the index is summed over with an appropriate metric tensor to raise one of the indices in the square. This is a shorthand that is more likely to be confusing and is used much less frequently than the Einstein summation convention itself.
  4. Jul 28, 2015 #3


    User Avatar
    Gold Member

    As you should know, ## \partial_\mu \phi## is a covariant first rank tensor, so you may name it ##S_\mu##. Now considering ##S^2\equiv S_\mu S^\mu##, will give you what you want.
  5. Jul 29, 2015 #4


    User Avatar
    Staff Emeritus
    Science Advisor

    Specifically, that is the Einstein summation convention for tensors, a notational convention: The same index, both as a subscript and a superscript is interpreted as a summation index.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Expanding field derivative
  1. Expanding log(a+b) (Replies: 8)

  2. Expand and factorise (Replies: 3)