# What is Tensor notation: Definition and 56 Discussions

This is a glossary of tensor theory. For expositions of tensor theory from different points of view, see:

Tensor
Tensor (intrinsic definition)
Application of tensor theory in engineering scienceFor some history of the abstract theory see also Multilinear algebra.

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1. ### I When can I commute the 4-gradient and the "space-time" integral?

Let's say I have the following situation $$I = \dfrac{\partial}{\partial x^{\alpha}}\int e^{k_{\mu}x^{\mu}} \;d^4k$$ Would I be able to commute the integral and the partial derivative? If so, why is that? In the same line of thought, in the situation I'm able to commute, would the result of...
2. ### I Terminologies used to describe tensor product of vector spaces

Hi, I'm in trouble with the different terminologies used for tensor product of two vectors. Namely a dyadic tensor product of vectors ##u, v \in V## is written as ##u \otimes v##. It is basically a bi-linear map defined on the cartesian product ##V^* \times V^* \rightarrow \mathbb R##. From a...
3. ### Engineering Tensor form of linear Hooke's law with E and v

Actually, this is not homework, but I think I need help like homework. It was raised from the notice that there is no tensor form of linear Hooke's law in terms of Young's modulus E, and Poission's ratio, v. For example, if we use lame parameters, we have G, \lambda, like The linear Hooke's...
4. ### B Array Representation Of A General Tensor Question

So, I've been watching eigenchris's video series "Tensors for Beginners" on YouTube. I am currently on video 14. I, in the position of a complete beginner, am taking notes on it, and I just wanted to make sure I wasn't misinterpreting anything. At about 5:50, he states that "The array for Q is...
5. ### B Transformation Rules For A General Tensor M

So, I've been watching eigenchris's video series "Tensors for Beginners" on YouTube. I am currently on video 14. I am a complete beginner and just want some clarification on if I'm truly understanding the material. Basically, is everything below this correct? In summary of the derivation of the...
6. ### B Beginner Einstein Notation Question On Summation In Regards To Index

So, I have recently been trying to learn how to work with tensors. In doing this, I have come across Einstein Notation. Below is my question. $$(a_i x_i)_{e}= (\sum_{i=1}^3 a_i x_i)_r=(a_1 x_1+a_2 x_2+a_3 x_3)_r$$; note that the following expression is in three dimensions, and I use the...
7. ### I Usage of First Order Elastic Constants in Soft Body Equations

Hi, I have some soft body equations that require first order elasticity constants. Just trying to figure out the proper indexing. From Finite Elements of Nonlinear Continua by J.T. Oden, the elastic constants I am trying to obtain are the first order, circled below: My particular constitutive...
8. ### I A little clarification on Cartesian tensor notation

Goldstein pg 192, 2 edIn a Cartesian three-dimensional space, a tensor ##\mathrm{T}## of the ##N## th rank may be defined for our purposes as a quantity having ##3^{N}## components ##T_{i j k}##.. (with ##N## indices) that transform under an orthogonal transformation of coordinates...
9. ### I Purpose of Tensors, Indices in Tensor Calculus Explained

I would like to know what is the utility or purpose for which the elements below were defined in the Tensor Calculus. They are things that I think I understand how they work, but whose purpose I do not see clearly, so I would appreciate if someone could give me some clue about it. Tensors. As...
10. ### I Expressing Vectors of Dual Basis w/Metric Tensor

I'm trying to understand why it is possible to express vectors ##\mathbf{e}^i## of the dual basis in terms of the vectors ##\mathbf{e}_j## of the original basis through the dual metric tensor ##g^{ij}##, and vice versa, in these ways: ##\mathbf{e}^i=g^{ij}\mathbf{e}_j##...

32. ### Tensor Notation for Triple Scalar Product Squared

Homework Statement Hi all, Here's the problem: Prove, in tensor notation, that the triple scalar product of (A x B), (B x C), and (C x A), is equal to the square of the triple scalar product of A, B, and C. Homework Equations The Attempt at a Solution I started by looking at the triple...
33. ### Magnetic field from vector potential function using tensor notation

Homework Statement We will see (in Chap. 5) that the magnetic field can be derived from a vector potential function as follows: B = ∇×A Show that, in the special case of a uniform magnetic field B_{0} , one possible vector potential function is A = \frac{1}{2}B_{0}×r MUST USE TENSOR NOTATIONm...
34. ### MHB Tensor notation for vector product proofs

I am new to tensor notation, but have known how to work with vector calculus for a while now. I understand for the most part how the Levi-Civita and Kronecker Delta symbol work with Einstein summation convention. However there are a few things I'm iffy about. For example, I have a problem where...
35. ### Confusion with Einstein tensor notation

Homework Statement I'm confused about writing down the equation: \Lambda \eta \Lambda^{-1} = \eta in the Einstein convention. Homework Equations The answer is: \eta_{\mu\nu}\Lambda^{\mu}{}_{\rho}\Lambda^{\nu}{}_{\sigma} = \eta_{\rho\sigma} However it's strange because there seems...
36. ### Commutator with Tensor Notation

Greetings, I would like to find the commutator \left[Lx^2,Ly^2\right] and prove that \left[Lx^2,Ly^2\right]=\left[Ly^2,Lz^2\right]=\left[Lz^2,Lx^2\right] I infer from the cyclic appearance of the indices that using the index notation would be much more compact and insightful to solve the...
37. ### Prove Perpendicularity of (AxB) and A Using Tensor Notation

Homework Statement Prove that (AxB) is perpendicular to A *We know that it is in the definition but this requires an actual proof. This is what I did on the exam because it was quicker than writing out the vectors and crossing and dotting them. Homework Equations X dot Y = 0 when...
38. ### Maxwell Equations in Tensor Notation

2A\mu=-\muoJ\mu Griffith's Introduction to Electrodynamics refers to this 4-vector equation as "the most elegant (and the simplest) formulation of Maxwell's equations." But does this encapsulate the homogeneous Maxwell Equations? I see how the temporal components lead to Gauss' Law, and I'm...
39. ### What is the constant C for Hodge dual in tensor notation?

So I know that the Hodge dual of a p-form A_{\mu_1 \mu_2\cdot \cdot \cdot \mu_p} in d dimensions is given by (*A)^{\nu_1 \nu_2 \cdot \cdot \cdot \nu_{d-p}} = C\epsilon^{\nu_1 \nu_2 \cdot \cdot \cdot \nu_{d-p}\mu_1 \mu_2 \cdot \cdot \cdot \mu_p}A_{\mu_1 \mu_2\cdot \cdot \cdot \mu_p} where C...
40. ### Wedge product in tensor notation

Is the following the definition of wedge product in tensor notation? Let A \equiv A_i be a matrix one form. Then A \wedge A \wedge A \wedge A \wedge A = \epsilon^{abcde}A_a A_b A_c A_d A_e ? in 5 dimensions. This question is in reference to the winding number of maps.
41. ### Question on tensor notation in group theory

in the appendix on Group Theory in Zee's book there is a discussion of commutations for SO(3) two questions - does [J^{ij},J^{lk}] = J^{ij}*J^{lk}-J^{lk}*J^{ij}? and there is an expression in the appendix that the commutator equals i(\delta^{ik}J^{jl} ... i don't understand the why...
42. ### What Do Upper and Lower Indices in Tensor Notation Signify?

Hi, I am very new to general relativity and have only just started to learn how to do some very basic manipulation of tensors. I can understand the methods I am using and have some idea of what a tensor is but am not sure what the difference between upper and lower indices signifies. I can...
43. ### What is the significance of square brackets in tensor notation?

What is meant by things like: R_{[abc]} and also things like: \nabla_{[a\nabla_b]} Where you have square brackets in the subscript? Thx
44. ### Help with tensor notation and curl

Homework Statement Show that \nabla \times (a \cdot \nabla a) = a\cdot\nabla(\nabla\times a) + (\nabla \cdot a(\nabla \times a) - (\nabla \times a)\cdot \nabla aHomework Equations \nabla \times (\nabla \phi) = 0 \nabla \cdot (\nabla \times a) = 0 The Attempt at a Solution I started with...
45. ### Solving Tensor Notation Issue Homework

Homework Statement I'm looking at the wikipedia article about four-momentum and I can't seem to get things right. It says Calculating the Minkowski norm of the four-momentum gives a Lorentz invariant quantity equal (up to factors of the ''c'') to the square of the particle's proper mass...
46. ### How Does Tensor Notation Work in Group Theory Calculations?

1. While reading notes on group theory there is a step I could not reproduce although it seems to me it should be straightforward. Probably there is something I am missing on tensor indices notation. Since R is an orthogonal matrix you can... 2 ...go from \epsilon...
47. ### How Do You Convert Complex Tensor Notation to Vector Notation?

Hi, I have the following term in tensor notation \frac{\partial{c}}{\partial{x_i}}\frac{\partial{u_i}}{\partial{x_j}}\frac{\partial{c}}{\partial{x_j}} I'm not sure how to write this in vector notation. Would it be? \nabla{c}\cdot\nabla\boldsymbol{u}\cdot{c} The problem I have...
48. ### Proving vector identities using Cartesian tensor notation

Homework Statement 1. Establish the vector identity \nabla . (B x A) = (\nabla x A).B - A.(\nabla x B) 2. Calculate the partial derivative with respect to x_{k} of the quadratic form A_{rs}x_{r}x_{s} with the A_{rs} all constant, i.e. calculate A_{rs}x_{r}x_{s,k} Homework Equations The...
49. ### Diagrammatic Tensor Notation from the Beginning

I've posted this in the Geometry & Topology section, but I believe it will get many more views here, so I'm posting a link Pictures here: https://www.physicsforums.com/showthread.php?t=407776 --- I really liked Penrose's diagrammatic way of writing tensor algebra, so I spent a while...
50. ### Diagrammatic Tensor Notation from the Beginning

I really liked Penrose's diagrammatic way of writing Tensor algebra, so I spent a while learning the basic notation. Unfortunately, it took a very long time for me to learn this because there is so little info on it to begin with. I also didn't see much mention of how to use the notation for...