Intuition for Covariant derivative of a Tensor Field

In summary, the covariant derivative of a tensor field is a way to measure how the vector changes as you move in a particular direction. The tensor covariant derivative can be calculated and proven to be useful in various situations.
  • #1
maze
662
4
Hi. I am attempting to gain some intuition for what the covariant derivative of a tensor field is.

I have a good intuition about the covariant derivative of vector fields (measuring how the vector changes as you move in a particular direction), and I understand how to extend the covariant derivative to tensors with the leibnitz product rule. I can calculate the tensor covariant derivative and prove things about it, but I still don't understand what it *is*.

Anyone?
 
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  • #2
Basic Tensor Understanding

There is a site at NASA, that is a introduction to tensor analysis. It starts with scalars, vectors, dyads ...etc. It is an excellent paper to understand what is really going on with tensors, and its free. It also has co. and contra. variant tensors.

What a self promoting site. I cannot post the URL, because I have to make 15 posts or more. I am just trying to help someone. Thats what happens when a site starts to go commercial (Sci. Amer.). Contact me and I will e-mail the address.
 
  • #3
well, you can't blame a site which ends with .com of being commercial.
 
  • #4
I googled NASA tensor analysis, is this it:
gltrs.grc.nasa.gov/reports/2005/TP-2005-213115.pdf

?
 
  • #5
I'm reading this pdf and so far it is rocking hard. nice.
 
  • #6
I sent you the whole name. It is a little different, but you might have the same one, or a variant.
 
  • #7
Thanks.
 
  • #8
Your welcome. It is kind of like all the nuts and bolts you cannot get unless you get a really great teacher. It is real basic stuff, but I promise you will learn tensors and really understand what is going on at the root level.
 
  • #9
I hope so. Right now working with tensors seems like a game you play by manipulating symbols according to random rules, although that is changing somewhat as I am reading the first pdf. After that, I'll get to the second pdf, and hopefully I'll be good after that.
 
  • #10
tensors

Forgot to tell you. After you read the pdf from nasa, get the schaums outline "Tensor Calculus", after those two things you will have no problem following any tensor book.

GOOD LUCK !
 

FAQ: Intuition for Covariant derivative of a Tensor Field

What is a covariant derivative of a tensor field?

A covariant derivative of a tensor field is a mathematical operation that measures the rate of change of the tensor field along a specified direction. It takes into account the curvature and coordinate system of the space in which the tensor field is defined.

How is the covariant derivative different from the regular derivative?

The covariant derivative takes into account the curvature of the space, while the regular derivative does not. This means that the covariant derivative is a more general form of the derivative, applicable to curved spaces, while the regular derivative is only applicable to flat spaces.

Why is the covariant derivative important in physics?

The covariant derivative is important in physics because it is used to describe the behavior of physical quantities on curved manifolds, such as in general relativity. It allows us to properly account for the effects of curvature and coordinate transformations in our calculations.

What is the mathematical notation for the covariant derivative?

The mathematical notation for the covariant derivative is ∇aTbc, where ∇ is the covariant derivative operator, a represents the direction of differentiation, and Tbc is the tensor field being differentiated.

How is the covariant derivative calculated?

The covariant derivative is calculated using the Christoffel symbols, which represent the connection coefficients between coordinate systems in a curved space. These symbols are used to modify the regular derivative in order to take into account the curvature of the space and calculate the covariant derivative.

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