Podcast #128: We chat with Kent C Dodds about why he loves React and discuss what life was like in the dark days before Git. Listen now.

Questions tagged [tensor-calculus]

Tensor calculus (tensor analysis) is a systematic extension of vector calculus to multivector and tensor fields in a form that is independent of the choice of coordinates on the relevant manifold, but which accounts for respective sub-spaces, their symmetries, and their connections.

Filter by
Sorted by
Tagged with
46
votes
9answers
13k views

What is a tensor?

I have a pretty good knowledge of physics, but couldn't deeply understand what a tensor is and why it is so fundamental.
37
votes
8answers
5k views

Is it foolish to distinguish between covariant and contravariant vectors?

A vector space is a set whose elements satisfy certain axioms. Now there are physical entities that satisfy these properties, which may not be arrows. A co-ordinate transformation is linear map from a ...
24
votes
1answer
4k views

Mathematically, what is color charge?

A similar question was asked here, but the answer didn't address the following, at least not in a way that I could understand. Electric charge is simple - it's just a real scalar quantity. Ignoring ...
43
votes
4answers
12k views

What is the physical meaning of the connection and the curvature tensor?

Regarding general relativity: What is the physical meaning of the Christoffel symbol ($\Gamma^i_{\ jk}$)? What are the (preferably physical) differences between the Riemann curvature tensor ($R^i_{\ ...
56
votes
7answers
30k views

Are matrices and second rank tensors the same thing?

Tensors are mathematical objects that are needed in physics to define certain quantities. I have a couple of questions regarding them that need to be clarified: Are matrices and second rank tensors ...
28
votes
4answers
11k views

What is the difference between a spinor and a vector or a tensor?

Why do we call a 1/2 spin particle satisfying the Dirac equation a spinor, and not a vector or a tensor?
9
votes
3answers
1k views

Staggered Indices ($\Lambda^\mu{}_\nu$ vs. $\Lambda_\mu{}^\nu$) on Lorentz Transformations

I have some open-ended questions on the use of staggered indices in writing Lorentz transformations and their inverses and transposes. What are the respective meanings of $\Lambda^\mu{}_\nu$ as ...
25
votes
7answers
3k views

Tensor Operators

Motivation. I was recently reviewing the section 3.10 in Sakurai's quantum mechanics in which he discusses tensor operators, and I was left desiring a more mathematically general/precise discussion. ...
27
votes
3answers
26k views

Why is stress a tensor quantity?

Why is stress a tensor quantity? Why is pressure not a tensor? According to what I know pressure is an internal force whereas stress is external so how are both quantities not tensors? I am ...
32
votes
5answers
22k views

Why is the covariant derivative of the metric tensor zero?

I've consulted several books for the explanation of why $$\nabla _{\mu}g_{\alpha \beta} = 0,$$ and hence derive the relation between metric tensor and affine connection $\Gamma ^{\sigma}_{\mu \beta}...
12
votes
5answers
2k views

Motivation for tensor product in Physics

This question is about a mathematical object (the tensor product) but thinking about the motivation that comes from Physics. Algebraists motivate the tensor product like that: "given $k$ vector spaces ...
17
votes
4answers
804 views

Difference between matrix representations of tensors and $\delta^{i}_{j}$ and $\delta_{ij}$?

My question basically is, is Kronecker delta $\delta_{ij}$ or $\delta^{i}_{j}$. Many tensor calculus books (including the one which I use) state it to be the latter, whereas I have also read many ...
8
votes
3answers
4k views

How to understand the definition of vector and tensor?

Physics texts like to define vector as something that transform like a vector and tensor as something that transform like a tensor, which is different from the definition in math books. I am having ...
3
votes
3answers
2k views

Square bracket notation for anti-symmetric part of a tensor

I know that $A_{[a} B_{b]} = \frac{1}{2!}(A_{a}B_{b} - A_{b}B_{a})$ But how can write $E_{[a} F_{bc]}$ like the above? Can you provide a reference where this notational matter is discussed?
3
votes
1answer
969 views

Transpose of (1,1) tensor

When we transpose a (1,1) tensor, shall we simply switch the two indices while keeping their upper/lower positions or switch them and also switch their upper/lower positions? In general, would the ...
5
votes
1answer
503 views

Expanding electromagnetic field Lagrangian in terms of gauge field

The electromagnetic field tensor is given by $F_{\mu \nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}$, and it appears in the Lagrangian as $$L = -\frac{1}{4}F_{\mu\nu}^2 - A_{\mu}J_{\mu}.$$ ...
23
votes
2answers
3k views

Introduction to spinors in physics, and their relation to representations

First, I shall say that I am familiar with the intuitive idea that a spinor is like a vector (or tensor) that only transforms "up to a sign" when acted on by the rotation group. I have even rotated a ...
1
vote
3answers
884 views

Convention of tensor indices

Let $g_{ij}$ be the diagonal Minkowski metric tensor diag$(g) = (1,-1,-1,-1)$, then $g^{ij}$ is defined to be $(g^{-1})^{ij}$, hence $$g_{ik}g^{kj} = g_i^{\ \ j} = \text{diag}(1,1,1,1)=\delta_i^{\ \ j}...
4
votes
1answer
1k views

$SU(3)$ irreducible representations with tensor method

I am dealing with the tensor product representation of $SU(3)$ and I have some problems in understanding some decomposition. 1) Let's find the irreducible representation of $3\otimes\bar{3}$ we have ...
6
votes
1answer
1k views

Difference between Cartesian product and tensor product on gauge groups

After a comment of John Baez to a question I asked on MathOverflow, I would like to ask what the difference between, for example, $SU(3)\times SU(2) \times U(1) $ and $SU(3) \otimes SU(2) \otimes U(1)$...
8
votes
1answer
1k views

How to get result $3 \otimes 3 = 6 \oplus \bar{3}$ for $SU(3)$ irreducible representations?

Let's have $SU(3)$ irreducible representations $3, \bar{3}$. How to get result that $$ 3\otimes 3 =6 \oplus \bar{3}~? $$ I'm interested in $\bar{3}$ part. It's clear that for $3 \otimes 3$ we can use ...
7
votes
4answers
2k views

Open source computer algebra systems for general relativity

I would like to use an open source computer algebra system (CAS) for research in general relativity. Kindly suggest a good choice between Ipython and Sage. I am more interested in the theoretical ...
6
votes
1answer
720 views

Physical interpretation of 2-forms dual to pseudovectors

Mathematically for every 3D pseudovector $x^i$ there is a 2-form $F_{ij}=\epsilon_{ijk}x^k$ such that the 2-form transforms properly under all orthogonal transformations. Therefore I would expect it ...
5
votes
2answers
509 views

“Vectors” (i.e. 1-tensors) their definition and motivation for relativity

I'm reading Einstein Gravity in a Nutshell (by Zee) and here he defines a vector as an object which is invariant under coordinate representation; concretely, if in one coordinate representation, $V$, $...
8
votes
3answers
2k views

Covariant and contravariant vectors

Reading Weinberg's Gravitation and Cosmology, I came across the sentence (p.115, above equation (4.11.8)) The partial derivative operator $\partial/\partial x^\mu$ is a covariant vector, or in ...
2
votes
1answer
496 views

Why is the Mixed Faraday Tensor a matrix in the algebra so(1,3)?

The mixed Faraday tensor $F^\mu{}_\nu$ explicitly in natural units is: $$(F^\mu{}_\nu)=\left(\begin{array}{cccc}0&E_x&E_y&E_z\\E_x&0&B_z&-B_y\\E_y&-B_z&0&B_x\\E_z&...
13
votes
2answers
3k views

Invariance, covariance and symmetry

Though often heard, often read, often felt being overused, I wonder what are the precise definitions of invariance and covariance. Could you please give me an example from quantum field theory?
13
votes
3answers
2k views

How to prove the covariant derivative cannot be written as an eigendecomposition of the partial derivative?

The Question How does one prove that Rindler's definition of the covariant derivative of a covariant vector field $\lambda_a$ as \begin{align} \lambda_{a;c} = \lambda_{a,c} - \Gamma^{b}_{\ \ ca} \...
8
votes
2answers
2k views

Understanding the difference between co- and contra-variant vectors

I am looking at the 4-vector treatment of special relativity, but I have had no formal training in Tensor algebra and thus am having difficulty understanding some of the concepts which appear. One ...
5
votes
3answers
527 views

Symmetry in terms of matrices

When we encounter a problem in physics which can be expressed in terms of matrices or tensors, why do we decompose the tensor in terms of its symmetric and antisymmetric or trace components? What is ...
4
votes
1answer
500 views

Working with indices of tensors in special relativity

I'm trying to understand tensor notation and working with indices in special relativity. I use a book for this purpose in which $\eta_{\mu\nu}=\eta^{\mu\nu}$ is used for the metric tensor and a vector ...
1
vote
1answer
297 views

How to compute the inertia tensor ${\bf{J}} _{\Omega}$ of a body of revolution

Suppose that $\Omega$ is a body of revolution of the function $y=f(x), a\le x \le b$ around the $x$-axis, where $f(x)>0$ is continuous. How to compute the inertia tensor ${\bf{J}} _{\Omega}$? ...
1
vote
2answers
713 views

Advection operator

How are exactly $u_j\partial_ju_i$ and $u_i\partial_j u_i$ related? And what is their relation to ($\boldsymbol{u}\cdot\nabla)\boldsymbol{u}$ and $\boldsymbol{u}\cdot(\nabla\boldsymbol{u})$ ? I ask ...
29
votes
4answers
8k views

Irreducible tensors concept

This might be a little naive question, but I am having difficulty grasping the concept of irreducible tensors. Particularly, why do we decompose tensors into symmetric and anti-symmetric parts? I have ...
11
votes
2answers
3k views

The Spin Connection

Why do we need to introduce the spin connection coefficients $\omega_{\mu \space \space b}^{\space \space a} $ in General Relativity? To me, they just look (mathematically) like the Christoffel ...
11
votes
2answers
2k views

Symmetry of the $3\times 3$ Cauchy Stress Tensor

When presenting the stress tensor (say in a non-relativistic context), it is shown to be a tensor in the sense that it is a linear vector transformation: it operates on a vector $n$ (the normal to a ...
8
votes
7answers
1k views

Is there a physical interpretation of a tensor as a vector with additional qualities?

What is a tensor? has been asked before, with the most highly up-voted answer defining a tensor of rank $k$ as a vector of a tensor of rank $k-1$. But if a scalar is defined as a physical quantity ...
6
votes
3answers
2k views

How to visualize the gradient as a one-form?

I am reading Sean Carrol's book on General Relativity, and I just finished reading the proof that the gradient is a covariant vector or a one-form, but I am having a difficult time visualizing this. I ...
6
votes
1answer
334 views

On the Lorentz Group representation [closed]

I am going through the notes on QFT by Srednicki. When describing fermions, from the very beginning he introduces the Lorentz Group and its algebra, and proves that it is equivalent to two copies of $...
5
votes
2answers
1k views

Covariance of Euler-Lagrange equations under change of generalized coordinates

Suppose I have an inertial frame with coordinate $\{q\}$. Now I define another reference frame with coordinate $\{q'(q,\dot q,t)\}$. I obtain the equation of motion in $\{q'\}$ in two different ways: ...
3
votes
1answer
966 views

How to write a generic density matrix for multi qubit system

I was reading the paper device independent outlook on quantum mechanics. The author defines a generic two qubit density matrix as $$ \rho=\frac{1}{4}\left( I \otimes I + \vec{r_{\rho}} \cdot \vec{\...
3
votes
2answers
1k views

What does it mean that the electromagnetic tensor is anti-symmetric?

The Faraday electromagnetic tensor is anti-symmetric. What is the physical meaning of this anti-symmetry?
7
votes
1answer
448 views

Non-Euclidean Phase Space?

In classical mechanics, the canonical equations of motion can be rendered in terms of Poisson Brackets: $$\begin{align} \left\{q_i, F(\mathbf{q},\mathbf{p})\right\} &= \frac{\partial F}{\partial ...
2
votes
2answers
307 views

Can Dirac equation be reformulated in an equivalent tensor form?

1) Can Dirac equation (including bispinors) be represented by a tensor formalism? 2) If yes, what kind of tensors could be the components of the wave function in Dirac equation in such formulation? ...
2
votes
1answer
984 views

How to check if a tensor transform a tensor?

Suppose $A^{\mu_1 \cdots \mu_n}_{\nu_{n+1}\cdots \nu_m}$ is a tensor. That means it transforms a tensor. How do I show that it transforms as a tensor? How do I see that $\cos (A^{\mu_1 \cdots \mu_n}_{\...
2
votes
2answers
5k views

Kronecker delta confusion

I'm confused about the Kronecker delta. In the context of four-dimensional spacetime, multiplying the metric tensor by its inverse, I've seen (where the upstairs and downstairs indices are the same): ...
14
votes
4answers
3k views

Is force a contravariant vector or a covariant vector (or either)?

I don't understand whether something physical, like velocity for example, has a single correct classification as either a contravariant vector or a covariant vector. I have seen texts indicate that ...
20
votes
2answers
3k views

Can any rank tensor be decomposed into symmetric and anti-symmetric parts?

I know that rank 2 tensors can be decomposed as such. But I would like to know if this is possible for any rank tensors?
18
votes
7answers
3k views

Why do we select the metric tensor for raising and lowering indices?

Any rank 2 tensor would do the same trick, right? Then, which is the motivation for choosing the metric one? Also, if you help me to prove that $g^{kp}g_{ip}=\delta^k_i$, I would be thankful too ^^...
15
votes
1answer
871 views

What are the local covariant tensors one can form from the metric?

Normally in differential geometry, we assume that the only way to produce a tensorial quantity by differentiation is to (1) start with a tensor, and then (2) apply a covariant derivative (not a plain ...