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I am currently a 3rd year undergraduate electronic engineering student. I have completed a course in dynamics, calculus I, calculus II and calculus III.

I've started self studying tensor calculus, my sources are the video lecture series on the YouTube channel; "MathTheBeautiful" and the freeware textbook/notes; "Introduction to Tensor Calculus" by Kees Dullemond & Kasper Peeters. Other textbooks go much more in depth in advanced math topics. I have been through the first 3 chapters and watched the first 5 videos, but I don't seem to understand the content. I don't know what I should take from these lectures and notes and what part of the work to focus on in order to start practicing as soon as possible.

I want to learn tensor calculus in order to study more advanced mathematics and physics such as; General Relativity, Differential Geometry, Continuum Mechanics etc. I've also seen many other textbooks on continuum mechanics and tensor analysis for mathematicians/physicists. All of these sources seem quite different and seem like I require much more advanced topics in mathematics in order to understand. How should I approach tensor calculus? through a physics or through a mathematics perspective?

From what I've seen, tensor calculus seems very abstract and more towards the proving side of the spectrum (like a pure mathematics subject), it doesn't look "practicable" as appose to other calculus courses where I could go to any chapter in the textbook and find many problems to practice and become familiar with the concept.

Is my current knowledge in calculus and physics + dynamics enough, or do I need to first learn a few more concepts in mathematics in order to begin attacking tensor calculus problems?

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DROP EVERYTHING AND GO STUDY LINEAR ALGEBRA

If you haven't taken an advanced linear algebra class, dealing not just with matrices and row reduction, but with vectors, bases, and linear maps, do that. The key to understanding tensor calculus at a deep level begins with understanding linear and multilinear functions between vector spaces. Once linear maps, multilinear maps, tensor products of spaces, etc., are clear to you, come back to this answer.

(Also, as a bonus, deeply understanding linear algebra will also make you understand calculus much better as well.)


Have you studied linear algebra now? Good. 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. Given a vector $v$ at that point, the stress tensor $\sigma$ produces the stress vector acting on the plane perpendicular to $v$ through that point. 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. I recommend Semi-Riemannian Geometry, with Applications to Relativity by Barrett O'Neill. (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.) You might start by working through his undergraduate curves & surfaces book, Elementary Differential Geometry.

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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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