Questions tagged [tensors]

For questions about tensors, tensor computation and specific tensors (e.g.curvature tensor, stress tensor). Tensor calculus is a technique that can be regarded as a follow-up on linear algebra. It is a generalisation of classical linear algebra. In classical linear algebra one deals with vectors and matrices.

Tensors, are arrays of numbers, or functions, that transform according to certain rules under a change of coordinates. In physics, tensors characterize the properties of a physical system.

A tensor may be defined at a single point or collection of isolated points of space (or space-time), or it may be defined over a continuum of points. In the latter case, the elements of the tensor are functions of position and the tensor forms what is called a tensor field. This just means that the tensor is defined at every point within a region of space (or space-time), rather than just at a point, or collection of isolated points.

Tensor is a geometric object that maps in a multi-linear manner geometric vectors, scalars, and other tensors to a resulting tensor. Vectors and scalars which are often used in elementary physics and engineering applications, are considered as the simplest tensors. Vectors from the dual space of the vector space, which supplies the geometric vectors, are also included as tensors.

Tensors were conceived in $1900$ by Tullio Levi-Civita and Gregorio Ricci-Curbastro, who continued the earlier work of Bernhard Riemann and Elwin Bruno Christoffel and others, as part of the absolute differential calculus. The concept enabled an alternative formulation of the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor.

Tensors are important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics (stress, elasticity, fluid mechanics, moment of inertia, ...), electrodynamics (electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, ...), or general relativity (stress–energy tensor, curvature tensor, ... ) and others.

References:

https://en.wikipedia.org/wiki/Tensor

http://www.ita.uni-heidelberg.de/~dullemond/lectures/tensor/tensor.pdf

https://www.grc.nasa.gov/www/k-12/Numbers/Math/documents/Tensors_TM2002211716.pdf

https://www.quora.com/What-is-a-tensor

What are the Differences Between a Matrix and a Tensor?

An Introduction to Tensors

3201 questions
1
vote
0 answers

About the "dot product" in the material derivative

Recently the material derivative was mentioned in my fluid dynamics class. The definition given is $$\operatorname{MD}=\partial_t +\mathbf{u}~\boldsymbol{\cdotp}\nabla$$ Where of course $\nabla$ is taken to be the covariant derivative using the Levi…
K.defaoite
  • 8,167
  • 4
  • 15
  • 32
1
vote
1 answer

How to find the second derivative of an expression in tensor form.

I would like to calculate $\Box\phi$ whereby $\phi = exp(ip_{\mu}x^{\mu})$ and $\Box = \partial_{\mu}\partial^{\mu}$ and whereby $\partial_{\mu} = \frac{\partial}{\partial x^{\mu}} $ and $\partial^{\mu} = \frac{\partial}{\partial…
DJA
  • 573
  • 2
  • 13
1
vote
1 answer

Transpose of a 3D Tensor

I'm finding it difficult to wrap my head around how the transpose operation works for Tensors of Rank 3 and above. Here's an example in PyTorch I was doing a transpose of tensors of rank 3 and according to transpose rule for rank 2 tensors which…
LostAtlas
  • 13
  • 4
1
vote
1 answer

$T \cdot W=0$ for every skew symmetric tensor $T$ implies that $T$ is a symmetric tensor

I'm trying to prove by myself the following easy fact. The context is continuum mechanics (in particular I'm using Gurtin's book): Let $T$ a tensor and $W%$ a skew-symmetric tensor. If $T \cdot W =0 $ for every $ W \in \text{Skw}$, then $T$ is…
1
vote
0 answers

Defining the inverse of a tensor via the adjugate tensor

My professor definied the adjugate of a tensor $\mathbf{t}\in T^{1}_{1}(E)$ (E is just a vector space of dimension n) by defining its components as…
Alex
  • 11
  • 1
1
vote
1 answer

calculation of divergence of a covariant vector

$(x,y)$ are rectangular Cartesian coordinates in a plane and $(\bar{x},\bar{y})$ are coordinates defined by $$x=\frac{\bar{x}^2+\bar{y}^2}{2} \ \ , \ \ y=\bar{x}\bar{y}$$ If $A_1,A_2$ are coordinates of a covariant vector in the $\bar{x},\bar{y}$…
am_11235...
  • 1,619
  • 1
  • 7
  • 10
1
vote
1 answer

Two Definitions of the Weyl Tensor

I'm reading "Textbook in Tensor Calculus and Differential Geometry" by Prasun Kumar Nayak and came across the Weyl tensor/projective curvature tensor $C_{kijl}$. The book states that $$C_{kijl}=R_{kijl}+\frac{1}{1-N}(g_{kj}R_{il}-g_{kl}R_{ij})…
1
vote
1 answer

Doubt on Double Covariant Derivatives

I simply want to calculate the Components of Riemann Tensor for a general connection from the definition given by: \begin{equation} \mathrm{Riem}(V,W,U) := \nabla_{V}\nabla_{W}U - \nabla_{W}\nabla_{V}U - \nabla_{[V,W]}U \end{equation} But I'm…
1
vote
1 answer

Interpretation and Usefulness of Direct Product

I'm currently reading A Brief on Tensor Analysis By J. Simmonds. It says, Let the projection of a vector $\mathbf{v}$ on a vector $\mathbf{u}$ be denoted and defined as $$\mathrm{Proj}_\mathbf{u}\mathbf{v}\equiv (\mathbf{v}\cdot…
Young Kindaichi
  • 476
  • 4
  • 9
1
vote
0 answers

Dot product followed by tensor product

I am currently learning on tensor calculus and having difficulty in understanding the following derivation. Suppose that $u = (u_1,u_2,u_3)^T$, one of the formula is $$ \begin{align*} \left(u \cdot \nabla u\right)\cdot u &=…
Nicolas H
  • 191
  • 5
1
vote
0 answers

Is $\nabla_Y(df) = d(\nabla_Yg)$ for $f=g$?

$\nabla_Y(df)$ is either a (1,1)-tensor or (0,1)-tensor (which one is it by the way?), and so is $d(\nabla_Yg)$. Are these two actually the same thing if $f=g$? $f$ and $g$ are both some functions on a smooth manifold, $\nabla_Y$ is the affine…
1
vote
1 answer

Understanding the effect of affine connection and exterior derivative on a Tensor field

If we have a differentiable manifold $M$ and a vector field $X$, with $d$ the exterior derivative and $\nabla$ an affine connection on $M$. If $f$ is a smooth function, what are the $(q,r)$ tensor field components for the following tensor fields: $$…
user860717
1
vote
0 answers

Relationship between $\mathcal{L}_X(d(df))$ and $d(\mathcal{L}_X(df))$?

I was thinking since $\mathcal{L}_X(d(df))$ = $\mathcal{L}_X(d(\omega))$ = $\mathcal{L}_X(g)$, where g is a (0,2)-tensor, and $d(\mathcal{L}_X(df)) = d(d\mathcal{L}_X(f)) = d(d(X[f])$ which is a two-form, so a (0,2)-tensor. I don't know a lot about…
1
vote
1 answer

Prove that if $\tau$ and $\sigma$ are two dyadic tensors then $\tau\cdot \sigma=\text{tr}(T\cdot S^T)$.

So using orthonormal bases, I proved that if $\tau$ and $\sigma$ are dyadic tensors then its inner product is the trace of the product of the matrices $T$ and $S$ of its image in $\hom(V,V)$ under a certain isomorphism. So since the product of a…
1
vote
1 answer

The formula of matrix calculus

Suppose $A \in \mathbb{R}^{p \times p}$ is a semi-positive defined symmetric matrix. Then $A^{1/2}$ is well-defined. Now I want to know does there exists a formula for $$\frac{\partial A^{1/2} }{\partial A} ?$$ Thanks so much! BTW, if we find an…
1 2 3
99
100