What is Tensor calculus: Definition and 99 Discussions

In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime).
Developed by Gregorio Ricci-Curbastro and his student Tullio Levi-Civita, it was used by Albert Einstein to develop his general theory of relativity. Unlike the infinitesimal calculus, tensor calculus allows presentation of physics equations in a form that is independent of the choice of coordinates on the manifold.
Tensor calculus has many applications in physics, engineering and computer science including elasticity, continuum mechanics, electromagnetism (see mathematical descriptions of the electromagnetic field), general relativity (see mathematics of general relativity), quantum field theory, and machine learning.

Working with a main proponent of the exterior calculus Elie Cartan, the influential geometer Shiing-Shen Chern summarizes the role of tensor calculus:In our subject of differential geometry, where you talk about manifolds, one difficulty is that the geometry is described by coordinates, but the coordinates do not have meaning. They are allowed to undergo transformation. And in order to handle this kind of situation, an important tool is the so-called tensor analysis, or Ricci calculus, which was new to mathematicians. In mathematics you have a function, you write down the function, you calculate, or you add, or you multiply, or you can differentiate. You have something very concrete. In geometry the geometric situation is described by numbers, but you can change your numbers arbitrarily. So to handle this, you need the Ricci calculus.

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  1. M

    A Tensor Calculus and Divergence

    Hi PF! I have a question on the dyadic product and the divergence of a tensor. I've never formally leaned this, although I'm sure it's published somewhere, but this is how I understand the operators. Can someone tell me if this is right or wrong? Let's say I have some vector ##\vec{V} = v_x i +...
  2. J

    A Covariant derivative definition in Wald

    I'm working through Wald's "General Relativity" right now. My questions are actually about the math, but I figure that a few of you that frequent this part of the forums may have read this book and so will be in a good position to answer my questions. I have two questions: 1) Wald first defines...
  3. W

    Derivation of the Christoffel symbol

    I'm reading Zee's Gravity book, can anyone help me understand the explanation on this part, I understand everything except the last part, he said to use (I.4.14) so that I could solve for the quantity shown in the image, what does he mean by that and how?
  4. W

    Transformation to locally flat coordinates

    I'm reading A. Zee's GR book and I'm in the section in which he is showing how to transform coordinates to be locally flat in a neighborhood of a point. He said that we can always choose our neighborhood to be locally flat for any space of any dimension D. "Look at how the metric transforms...
  5. W

    Curvature at the origin of a space as described by a metric

    Homework Statement This is a problem from A. Zee's book EInstein Gravity in a Nutshell, problem I.5.5 Consider the metric ##ds^2 = dr^2 + (rh(r))^2dθ^2## with θ and θ + 2π identified. For h(r) = 1, this is flat space. Let h(0) = 1. Show that the curvature at the origin is positive or negative...
  6. D

    Lie derivative of tensor field with respect to Lie bracket

    I'm trying to show that the lie derivative of a tensor field ##t## along a lie bracket ##[X,Y]## is given by \mathcal{L}_{[X,Y]}t=\mathcal{L}_{X}\mathcal{L}_{Y}t-\mathcal{L}_{Y}\mathcal{L}_{X}t but I'm not having much luck so far. I've tried expanding ##t## on a coordinate basis, such that...
  7. Mentz114

    Tensor Calculus Problem: Simplifying Terms with Index Exchange

    If you don't like indexes, look away now. I got these terms from a tensor calculus program as part of a the transformed F-P Lagrangian. \begin{align} {g}^{b a}\,{g}^{d e}\,{g}^{f c}\,{X}_{a,b c}\,{X}_{d,e f}\\ +{g}^{b a}\,{g}^{c f}\,{g}^{e d}\,{X}_{a,b c}\,{X}_{d,e f}\\ +{g}^{b a}\,{g}^{c...
  8. U

    Lowering Indices: Tensor Calculus Basics

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  9. U

    Contracting \mu & \alpha - What Does It Mean?

    What do they mean by contracting ##\mu## with ##\alpha## ?
  10. U

    Energy-Momentum Tensor Algebra

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  11. U

    Einstein Tensor - Particle at rest?

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  12. U

    Flat Space - Christoffel symbols and Ricci = 0?

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  13. Abolaban

    Tensor calculus> definition of contravariants

    Hello Big minds, In the book of Arfken [Math Meth for Physicists] p 134 he defined contravariant tensor...my question is about a_ij he defined them first as cosines of an angle of basis then he suddenly replaced them by differential notation...why is that? cosines are not mention in this...
  14. F

    Understanding Covariant Derivative & Parallel Transport

    Hello, I try to apprehend the notion of covariant derivative. In order to undertsand better, here is a figure on which we are searching for express the difference \vec{V} = \vec{V}(M') - \vec{V}(M) : In order to evaluate this difference, we do a parallel transport of \vec{V}(M') at point...
  15. BiGyElLoWhAt

    Understanding Tensors for General Relativity: A Comprehensive Guide

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  16. J

    Calculation and application of dyadic Green's function

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  17. U

    Faraday Tensor and Index Notation

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  18. Telemachus

    Tensor calculus, dummy indices

    Hi there. When I have dummy indices in a tensor equation with separate terms, I wanted to know if I can rename the dummies in the separate terms. I have, in particular: \displaystyle w_k=-\frac{1}{4}\epsilon_{kpq}\left [ \frac{\partial u_p}{\partial x_q}-\frac{\partial u_q}{\partial x_p}...
  19. B

    Tensor calculus independent study questions?

    I'm a mathematics major and up until now I've taken Calc 1,2,3 (so single + multivariable) a combined course in Elementary Linear Algebra + Differential Equations and PDE's. My school doesn't offer any tensor calculus classes, but I was interested in learning some of it on my own. Do I have...
  20. aditya ver.2.0

    What is Tensor Calculus and How is it Related to Differential Geometry?

    I have started to learn a bit about Tensor calculus and it all going above my head. May anyone give a brief outline about the topic (preferably theoretical) and the supplementary concepts attached to it.
  21. aditya ver.2.0

    The modelling of space time through Riemann tensor calculus

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  22. aditya ver.2.0

    Understanding the Role of Tensor Calculus in General Relativity

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  23. C

    Why Can't We Do Algebraic Methods with Tensors?

    Hello everyone! Even though I have done substantial tensor calculus, I still don't get one thing. Probably I am being naive or even stupid here, but consider $$R_{\mu\nu} = 0$$. If I expand the Ricci tensor, I get $$g^{\sigma\rho} R_{\sigma\mu\rho\nu} = 0$$. Which, in normal algebra, should...
  24. C

    Derivations of Einstein field equations

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  25. putongren

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  26. Telemachus

    Tensor calculus, gradient of skew tensor

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  27. G

    What Are the Best Books for Mastering Tensor Calculus?

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  28. P

    Attempting to learn tensor calculus

    Hello all, After a brief break from attempting to learn tensor calculus, I'm once again back at it. Today, I started reading this: http://web.mit.edu/edbert/GR/gr1.pdf. I got to about page 4 before things stopped making sense, right under equation 3. Question 1: apparently a "one-form" is a...
  29. J

    Scalar, vector and tensor calculus

    I noticed that sometimes exist a parallel between scalar and vector calculus, for example: ##v=at+v_0## ##s=\int v dt = \frac{1}{2}at^2 + v_0 t + s_0## in terms of vector calculus ##\vec{v}=\vec{a}t+\vec{v}_0## ##\vec{s}=\int \vec{v} dt = \frac{1}{2}\vec{a}t^2 + \vec{v}_0 t + \vec{s}_0##...
  30. wavepart7cle

    Einstein's applications of tensor calculus

    Hey everyone, I recently learned that my certified genius weird-uncle-who-lives-at-home (IQ over 200 something, legitimate 'genius') or WULAH for short, passes his spare time by lounging around his place and doing tensor calculus. I've done some calc in 3d in college and I know that's commonly...
  31. A

    Tensor Calculus: Modern Books with Physics Examples

    Hello, could someone recommend a good book on tensor calculus? I'd like it to be relatively modern (I have an old book) and maybe contain some examples drawn from physics. Chapters on related subjects such as differential forms and calculus of variations would be a plus. Cheers.
  32. I

    Tensor calculus for general relativity question.

    Use the metic that Einstein proposed in the first cosmological model based on general relativity. ds2 = -dt2 + (dr2) / (1 - Kr2) + r2(dθ2 + sin2θd\phi2) where K > 0 Show that the stress energy tensor is that of a static, spatially uniform perfect fluid and determine ρ and p in terms of G and...
  33. J

    Trace - Integration - Average - Tensor Calculus

    Hi Let D be an anisotropic tensor. This means especially, that D is traceless. \mathrm{tr}(D) = 0 Apply the representating matrix of D to a basis vector S , get a new vector and multiply this by dot product to your basis vector. Than you got a scalar function. Now integrate this...
  34. A

    Is the Determinant of a Covariant Tensor of Order 2 an Invariant of Weight 2?

    Question: Let Aij denote an absolute covariant tensor of order 2. Show that the determinant A = det(Aij ) is an invariant of weight 2 and A is an invariant of weight 1. I have little clue about this question. Would writting down the transformation rule from barred to unbarred 2nd-order tensor...
  35. T

    Am I ready to learn tensor calculus? And what's a good book for it?

    Hi, the only thing I know is differential and integral calculus for functions with 1 variable and the basics of linear algebra (solving linear systems with matrices, determinants, etc...) . I'm currently learning differential and integral calculus for real functions with multiple real variables...
  36. C

    Help w/ Tensor Calc Homework: Bijk Properties Under Rotations

    Homework Statement if BijkAjk is a vector for all symmetric tensors Ajk, (but Bijk is not necessarily a tensor), what are the properties of Bijk under rotations of the basis/coordinate axes? Homework Equations The Attempt at a Solution I am not sure what the question is looking for... though I...
  37. S

    Tensor Calculus General Theory of Relativity

    Hello I have huge problems with the following exercise. Please give me some hints. No complete Solutions but a little bit help. Find the differential equations of the paths of test particles in the space-time of which the metric ist \mathrm{d}s^2 = e^{2kx} \left[- \left( \mathrm{d}x^2...
  38. X

    Any suggestions for understanding notation in Tensor Calculus?

    I have a book that I've been reading off and on: https://www.amazon.com/dp/012200681X/?tag=pfamazon01-20 It's enjoyable but the only issue I'm having with it is the notation used, still. The book is definitely good at what it wants to achieve, which is a bridging text to the math for...
  39. J

    Tensor Calculus Questions: Christoffel, Riemann & Ricci Tensors

    Dear Friends I have two questions to do about Tensor Calculus: 1) Is there any program to calculate Christoffel Symbols, Riemann and Ricci Tensors and everything about Tensor Calculus (Free or paid)? 2) When in an exercise anyone asks to use the Euclidean metric or Riemann metric, what...
  40. W

    Tensor calculus for general relativity

    I'm taking a course on relativity, both special and general. According to my college, I have the required mathematical background (vector analysis, electromagnetics (though I can't recall more than a cursory glance at tensors) etc) to make sense of it. Special relativity I can handle, and I...
  41. M

    The book Introduction to tensor calculus and continuum mechanics

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  42. J

    Complete set of answers to Schaum's Tensor Calculus

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  43. O

    Learning Tensor Calculus: Struggling and Need Help!

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  44. J

    Notational problem in tensor calculus

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  45. M

    Tensor Calculus: Ellipse Equation & Transformation to Cartesian Coordinates

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  46. Astronuc

    Introduction to Tensor Calculus and Continuum Mechanics

    Here is a free and downloadable textbook on Tensor Calculus. http://www.math.odu.edu/~jhh/counter2.html Scroll to bottom of page to find pdf files.
  47. P

    Help Solve a Mystery: Lawden's "An Introduction to Tensor Calculus

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  48. Q

    What is Tensor Calculus and Its Applications in Mechanics?

    Im only in second semester calculus and my friend keeps on babbling about Tensor Calculus and how only a few people know how to do it in the world. I highly doubt that only a few people in the world know how to do this because there are plenty of math graduates out there, as well as professors...
  49. H

    Differential geometry and tensor calculus

    i have been doing fourier, differential equations, and advanced calculus and then i saw differential geometry in a book...since teh book only covered advanced calculus, it only introduced diff. geometry...can anyone show and tell me where on the web there is a tutorial for differential geometry?
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