A Semicolon notation in component of covariant derivative

Summary
The use of semicolon notation as covariant derivative
Can someone clarify the use of semicolon in
245677

I know that semicolon can mean covariant derivative, here is it being used in the same way (is
245678
expandable?) Or is
245679
a compact notation solely for the components of
245680
?
 

pervect

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If you're familiar with index or abstract index notation

$$S^\alpha{}_{\beta\gamma;\delta} = \nabla_\delta \, S^{\alpha}{}_{\beta\gamma}$$
 
Can someone clarify the use of semicolon in
View attachment 245677
I know that semicolon can mean covariant derivative, here is it being used in the same way (is View attachment 245678 expandable?) Or isView attachment 245679 a compact notation solely for the components ofView attachment 245680?
Not sure what your confusion is. Any tensor can be written as the sum of each of its components in some basis multiplied by the outer product of the respective basis vectors and co-vectors. Since the covariant derivative of a tensor is still a tensor (but one rank higher), it can also be written this way. Does that clear anything up for you?
 
If you're familiar with index or abstract index notation

$$S^\alpha{}_{\beta\gamma;\delta} = \nabla_\delta \, S^{\alpha}{}_{\beta\gamma}$$
Hello thank you, can you tell me what did you use to type your equations? I'm new to this site and trying to find a good way to generate equations. Much appreciated.
 

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