- Principles of Tensor Calculus | Physics / Math / Brain Teasers | Calculus, Math notes, Mathematics
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Have you studied linear algebra now?
The intuition behind tensor calculus is that we can construct tensor fields smoothly varying from point to point. At every point of a manifold or Euclidean space, if you prefer we can conceptualize the vector space of velocities through that point. Once we have a vector space, we have its dual, and from the space and its dual, we construct all sorts of tensor spaces. A tensor field is just one such tensor at every point that varies in a differentiable fashion across the manifold. Let's make a concrete example.
You're an EE student, hopefully you'll forgive me if I use a concept from mechanical engineering. Consider a voluminous body with internal stresses. Fix a point.
Principles of Tensor Calculus | Physics / Math / Brain Teasers | Calculus, Math notes, Mathematics
It is a tensor because it does so in a linear fashion, at each point mapping a vector to another vector. If you're interested in general relativity and differential geometry, consider also picking up some differential geometry textbooks. As a plus, if by then your linear algebra is rusty, the first chapter is devoted to the basics of multilinear algebra and tensor mechanics.
First all, study multivariable differential calculus from Rudin's PMA. Then learn Smooth manifolds through Sinha's book, and Lee's book. Only then O'Neill's Semi-Reimannian geometry could be intelligible. This book will teach you the true SR and GR.
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