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Does anyone have any recommendations on good textbooks (or websites) that help translate the mathematicians' language of tensors (strictly as multilinear maps to the underlying field) and forms (as alternating tensor fields) to the language used by physicists?

For example, I want to know what covariance/contravariance is all about, in the context of multilinear algebra and analysis. Also, I still can't figure out how/why tensors can be visualized as an n-dimensional array of numbers.

I have found many sources that talk about the things above from first principles, but I find these to be un-illuminating. I am looking for a source that places these concepts in context to mathematical theory. (For example, I want to see a proof that shows that tensors can be n-dimensional arrays, and I want to see exactly how covariance/contravariance are related to pullbacks and push-forwards.)

I am a college undergraduate junior, and I have just finished working through Munkres' Analysis on Manifolds.

Thanks.

-maxx

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# Please help an ignorant mathematician

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