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Puzzler183

Covariant and contravariant tensors

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Usually, I'm a pretty math-smart guy; I'm only a few classes away from a math major. But lately, I've been reading (or attempting to read) about tensors and I can't figure out what the heck they mean by covariant and contravariant indices. Mathworld is no help (since it's more of a reference anyway) and wikipedia didn't do much good for me either... Can anyone explain clearly what they mean?

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A tensor T is usually followed by subscripts or superscripts a,b,c,....

If they are subscripts, then it represents a covariant tensor. If they are superscripts, then they are contravariant. It is possible to have a tensor with both covariant and contravariant components.

If you have a vector represented as a tensor T, then Ta is the conjugate transpose of Ta, for instance.

Tensors are a tricky subject, and they have an alien notation. I would recommend talking to one of your professors about them. You'll find even a large portion of them don't understand tensors. Maybe a physicist that is intimately familiar with electricity and magnetism, or relativity could help you. A math course that you could take that deals with tensors is differential geometry.

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Quote:
Original post by erissian
A tensor T is usually followed by subscripts or superscripts a,b,c,....

If they are subscripts, then it represents a covariant tensor. If they are superscripts, then they are contravariant. It is possible to have a tensor with both covariant and contravariant components.

If you have a vector represented as a tensor T, then Ta is the conjugate transpose of Ta, for instance.

Tensors are a tricky subject, and they have an alien notation. I would recommend talking to one of your professors about them. You'll find even a large portion of them don't understand tensors. Maybe a physicist that is intimately familiar with electricity and magnetism, or relativity could help you. A math course that you could take that deals with tensors is differential geometry.


Thank you! That part (the bold) was like ridiculously helpful. And yeah, I'm thinking I'm going to have to talk to one of my professors since these things are just confusing >_<.

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Assuming you have taken a class in general topology and some algebra, I highly recommend Boothby's "Introduction to Differentiable Manifolds and Riemannian Geometry" as an introduction to differential geometry. If you don't want to buy it, you can obtain a copy here

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Focusing on the superscript or subscript notation of covariant and contravariant vectors (I'm using the term vector in a general sense) is much like seeing trees without seeing the forest. The nature of covariant and contravariant vectors is really very simple. Their difference is in how they transform. The transformation of one coordinate set to another is generally a linear combination of the original coordinates. With covariant vectors the coefficients of the terms are the partial derivatives of the old components with respect to the new components. With contravariant vectors the coefficients of the terms are the partial derivatives of the new components with respect to the old components. It is really that "easy".

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