I Tensor Calculus (Einstein notation)

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Einstein notation can be confusing, particularly regarding the expression ∂uFv - ∂vFu, which is not necessarily zero for tensor Fu despite the indices running through the same values. The notation implies that Gμν = ∂μFν - ∂νFμ represents multiple equations, specifically 16 combinations for the indices μ and ν. The contraction of Gμν with φμ leads to the definition of ψν, which involves summing over the indices, clarifying the relationship between these terms. Understanding that Einstein notation omits explicit summation symbols helps in grasping the underlying mathematics. The discussion ultimately highlights the importance of recognizing how indices interact in tensor calculus.
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How to sum over indices when they aren't being contracted?
Hello,

I realize this might sound dumb, but I'm having such a hard time understanding Einstein notation. For something like ∂uFv - ∂vFu, why is this not necessarily 0 for tensor Fu? Since all these indices are running through the same values 0,1,2,3?
 
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Einstein notation for summation is meant to be done for the same term, here you have two terms. There is no implicit summation in ##\partial_\mu F_\nu - \partial_\nu F_\mu##.

Let's call it ##G_{\mu \nu}## i.e. ##G_{\mu \nu} = \partial_\mu F_\nu - \partial_\nu F_\mu##. We have
## G_{00} = \partial_0 F_0 - \partial_0 F_0 = 0##
## G_{10} = \partial_1 F_0 - \partial_0 F_1 = - G_{01}##
## G_{11} = \partial_1 F_1 - \partial_1 F_1 = 0 = G_{22} = G_{33}##
## G_{23} = \partial_2 F_3 - \partial_3 F_2 = - G_{32}##
and so on.

Now, lets assume I contract ##G_{\mu \nu}## with ##\phi^\mu##, we have due to Einstein summation convention
##\phi^\mu G_{\mu \nu} = \phi^0G_{0 \nu} + \phi^1G_{1\nu} + \phi^2G_{2 \nu} + \phi^3G_{3 \nu}##

Let's define ##\psi_\nu = \phi^\mu G_{\mu \nu}##.
We have
##\psi_0 = \phi^\mu G_{\mu 0} = \phi^0G_{0 0} + \phi^1G_{10} + \phi^2G_{20} + \phi^3G_{30} =\phi^1G_{10} + \phi^2G_{20} + \phi^3G_{30} ##
##\psi_1 = \phi^\mu G_{\mu 1} = \phi^0G_{0 1} + \phi^1G_{11} + \phi^2G_{21} + \phi^3G_{31} =\phi^0G_{0 1} + \phi^2G_{21} + \phi^3G_{31}##
and so on
 
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Ah I understand now, thank you!
 
paperplane said:
Ah I understand now, thank you!
I updated my reply above with some more examples.
 
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paperplane said:
TL;DR Summary: How to sum over indices when they aren't being contracted?

Hello,

I realize this might sound dumb, but I'm having such a hard time understanding Einstein notation. For something like ∂uFv - ∂vFu, why is this not necessarily 0 for tensor Fu? Since all these indices are running through the same values 0,1,2,3?
Einstein notation omits two standard elements of mathematical notation: the summation symbol (##\sum##) and the universal quantifier (##\forall##). In the above examples we have:
$$G_{\mu \nu} = \partial_\mu F_\nu - \partial_\nu F_\mu$$In full notation this would be:
$$\forall \mu, \nu: G_{\mu \nu} = \partial_\mu F_\nu - \partial_\nu F_\mu$$Note that this is actually ##16## equations! (One for every combination of ##\mu = 0, 1,2,3## and ##\nu = 0, 1,2,3##.) And:
$$\psi_\nu = \phi^\mu G_{\mu \nu}$$In full notation this would be:
$$\forall \nu: \psi_\nu = \sum_{\mu = 0}^{3} \phi^\mu G_{\mu \nu}$$And that is four equations.
 
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