I recommend Semi-Riemannian Geometry, with Applications to Relativity by Barrett O'Neill. If you're interested in general relativity and differential geometry, consider also picking up some differential geometry textbooks. It is a tensor because it does so in a linear fashion, at each point mapping a vector to another vector. 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. Consider a voluminous body with internal stresses. You're an EE student, hopefully you'll forgive me if I use a concept from mechanical engineering. A tensor field is just one such tensor at every point that varies in a differentiable fashion across the manifold. Once we have a vector space, we have its dual, and from the space and its dual, we construct all sorts of tensor spaces. At every point of a manifold (or Euclidean space, if you prefer) we can conceptualize the vector space of velocities through that point. The intuition behind tensor calculus is that we can construct tensor fields smoothly varying from point to point. Have you studied linear algebra now? Good. (Also, as a bonus, deeply understanding linear algebra will also make you understand calculus much better as well.) Once linear maps, multilinear maps, tensor products of spaces, etc., are clear to you, come back to this answer. The key to understanding tensor calculus at a deep level begins with understanding linear and multilinear functions between vector spaces. 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. 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?ĭROP EVERYTHING AND GO STUDY LINEAR ALGEBRA How should I approach tensor calculus? through a physics or through a mathematics perspective?įrom 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. All of these sources seem quite different and seem like I require much more advanced topics in mathematics in order to understand. I've also seen many other textbooks on continuum mechanics and tensor analysis for mathematicians/physicists. 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 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 have been through the first 3 chapters and watched the first 5 videos, but I don't seem to understand the content. Other textbooks go much more in depth in advanced math topics. 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
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