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.

2,254 questions
Filter by
Sorted by
Tagged with
107 views

Field of research treating tetrads (vierbeins) as fundamental objects?

After a lot of research on tetrads I think I found the subject I'd like to specialize in for postgrad/phd, as they seem to express many interesting and (perhaps) fundamental physical properties. So i ...
154 views

Proper conceptualization & notation for vectors, $n$-tuples, and matrices in physical space

This is a fairly basic question that I may be making longer than necessary. But it has plagued me for some time. It is essentially this: In what space do abstract physical vectors like a velocity ...
• 351
51 views

Advection term for a matrix equation

How can I calculate a quantity like $(\vec{v} \cdot \nabla) M$ where $\vec{v}$ is the velocity vector, and $M$ is some 3x3 matrix? (if one wants, assume $M$ is a tensor) This would be the advective ...
83 views

• 539
35 views

Can someone suggest me a book where whole Electrodynamics of master level is explained in tensor form

I am trying to study electrodynamics, can someone please suggest me a book or any other resource where I can get it in tensor form, Topics: waves in medium, resonators, etc.
54 views

Divergence theorem in index notation

From Batchelor's book of fluid dynamics: I guess that's an easy question for anyone having more familiartiy than me in tensor calculus, anyways. First integral argument is the i-component of the ...
1 vote
59 views

Lorentz transform of Levi-Civita Symbol

I was reading about Lorentz transformations and frequently I hear the notion of Lorentz transforming quantities like $\epsilon^{\mu \nu \rho \sigma}$. But I have never heard an explanation as to why ...
• 119
74 views

The displacement gradient tensor transformation rule

The transformation rule of a 2nd rank tensor expresssed in a given basis is often written as follow: $$F' = P^T FP$$ where $F$ is the matrix representation of the tensor in the old basis B, $F'$ its ...
144 views

How do we make sense of $F^{\mu\nu}F_{\mu\nu}$? The book just assumes I understand it

Why are these upper and lower indices and what does that mean. I can't interpret the term with upper indices.
• 119
26 views

Confusion on number of component of Cauchy stress tensor

The Cauchy stress tensor is often presented as a tensor having $(2,0)$ tensor having nine components in any given basis. However, I think it should actually be $6 \times 3 =18$ because a cube has six ...
99 views

What does $\delta/\delta t$-derivative represent in tensor calculus?

Some texts, such as Pavel Grinfeld's, talk about a $\delta/\delta t$-derivative whose role (in trajectory analysis of particles using tensor calculus) is pretty obscure to me. For example, the ...
• 300
32 views

How to calculate the electric field of a polarization density?

By polarization density here I just assume I have a "blob" of free positive and negative charges, and instead of describing the system with a charge density $\rho(\pmb{r})$ I want to use the ...
31 views

• 105
104 views

Expressing Maxwell's equations in tensor notation

I've been teaching myself relativity by reading Sean Carroll's intro to General Relativity textbook, and in the first chapter he discusses special relativity and introduces the concept of tensors, ...
• 105
16 views

Generalization of the Impact Depth Equation

Newtons Impact Depth Equation. $L = l *\dfrac{p1} {p2}$ $L$ is the impact depth $l$ is the length of the projectile $p1$ is the density of the projectile $p2$ is the density of the Target has many ...
• 101
67 views

Kerr Solution metric ansatz for EFEs

The Schwarzschild metric ansatz is given by $$ds^2=-A(r)dt^2+B(r)dr^2+r^2d \Omega^2$$ where upon applying the Einstein Field Equations $$G_{\mu\nu}=\kappa T_{\mu\nu}$$ we obtain the normal ...
• 335
43 views

Finding the proportionality constant in $\varepsilon^{\mu\nu}A_\mu^{\ \lambda} A_\nu^{\ \rho}\propto \varepsilon^{\lambda\rho}$

We can show that the contraction of some arbitrary $2\times2$ matrix $A_{\mu}^{\ \lambda}$ with the Levi-Civita symbol is once again antisymmetric \begin{align*} \varepsilon^{\mu\nu}A_\mu^{\ \lambda} ...
• 1,520
70 views

Show that the contraction of a covector and a vector is Lorentz invariant

I just got Sean Carroll's Spacetime and Geometry: An Introduction to General Relativity a couple of weeks ago, and I have resolved to go through the entire book. In the first chapter, he prompts the ...
• 105
27 views

Reference which explains Penrose Diagramatic notation in simple way

In both Penrose's Road to reality and Spinor's and space-time, the following notation is shown: With a lot other examples for doing calculation with Tensors. Could someone give another reference ...
227 views

What is the idea behind 2-spinor calculus?

In the book by Penrose & Rindler of "Spinors and Space-Time", the preface says that there is an alternative to differential geometry and tensor calculus techniques known as 2-spinor ...
86 views

• 11
41 views

Interpreting stress at the ends of a bar

Consider a bar loaded in tension by distributed loads applied on its ends as shown in the figure. The stress at any cross section of this bar will be $$\sigma = \frac{P}{A}$$ From what I know about ...
105 views

• 327
1 vote