Covariant derivatives

McGravity

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I understand covariant derivatives which are given in many textbooks, but is there such a thing as a partial covariant derivative, that is the covariant derivative(s) of Z as a function of x and y?
 

Subhotosh Khan

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I understand covariant derivatives which are given in many textbooks, but is there such a thing as a partial covariant derivative, that is the covariant derivative(s) of Z as a function of x and y?
Yes there is " partial covariant derivative".

Type in those terms in Google and you'll be referred to >1740 sites!
 

yoscar04

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Take a look at page 207, Gravitation by Misner, Thorne and Wheeler. This book is considered a classic.
 

yoscar04

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1594588343041.png
from Tensors, Differential forms and Variational Principles,
by David Lovelock, Hanno Rund.
 

Singleton

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What are you calling Z?
 

McGravity

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The actual function was V = r^2 cos(theta) and came in Relativity Demystified (ironic title!) by McMahon
 

McGravity

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That is a definition but I was hoping for an explanation, which isn't quite the same thing. If I can find how to covariantly differentiate V = r Cos theta that would be a great help.
 

yoscar04

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That is a definition but I was hoping for an explanation, which isn't quite the same thing. If I can find how to covariantly differentiate V = r Cos theta that would be a great help.
I thought that the covariant derivative for the case of a scalar function coincides with the gradient of it, and when you generalize it to include vectors and tensors you get also the connection coefficients. Am I wrong?
 

yoscar04

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I hope this will help you a bit more.
 

McGravity

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I thought that the covariant derivative for the case of a scalar function coincides with the gradient of it, and when you generalize it to include vectors and tensors you get also the connection coefficients. Am I wrong?
Yes that sounds right but it's just what you would find in any textbook. My problem came from Relativity Demystified (ironic title)
 

yoscar04

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No that's way above my head. I am a physicist/engineer, not a mathematician
Indeed it is heavy stuff, but standard differential geometry for GR.
 
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