Tagged Questions

Use this tag for questions about specific tensors (curvature tensor, stress tensor), or questions regarding tensor computations as they appear in multivariable calculus and differential/Riemannian geometry (specifically, when it is amenable to be treated as objects with multiple indices that ...

2answers
74 views

Why is the curvature tensor defined to be anti-symmetric?

I'm reading Chern's Lectures on Differential Geometry. On page 117 of section 4-2, he points out that \begin{eqnarray} d\omega_i^j - \omega_i^h \wedge\omega_h^j &=& \frac{\partial\Gamma^j_{ik}}...
1answer
64 views

Why does the Levi-Civita connection commute with pullbacks and pushforwards?

If $i: M \to N$ is an embedding of Riemannian manifolds, I am trying to prove that $\nabla i^* T = i^* \nabla T$ for any covariant tensor $T$ (I use the same letter for the two Levi-Civita connections)...
1answer
63 views

Calculating Christoffel symbols using variational geodesic equation

Given the line element $$ds^2 = e^v dt^2 - e^{\lambda} dr^2 - r^2 d \theta^2 - r^2 \sin^2 \theta d \phi^2$$ we wish to compute the Christoffel symbols $\Gamma^{a}_{bc}$ using the geodesic equation. ...
2answers
71 views

An example for a mathematical object that has two indexes, and is not a tensor

I understand how to check if something is a tensor (if it transforms like a tensor, it's a tensor). I have this example of a rank 2 tensor: $$\tau_{ij}=r_if_j - r_jf_i$$ and I understand that it is a ...
1answer
53 views

Divergence theorem in curvilinear coordinates

Suppose I have a tensor \begin{gather} \stackrel{\leftrightarrow}{A} = \begin{bmatrix} a_{11}(\vec{r}) & a_{12}(\vec{r}) & a_{13}(\vec{r})\\ a_{21}(\vec{r}) & a_{22}(\vec{r}) & a_{23}(...
1answer
41 views

Analyzing the Hessian of this function

The problem I'm looking at is, given a matrix $A$ of size $m\times n$ where $A_{ij}\ge 0$ minimize $f(x, y) = ||A - xy||^{2}_{F}$ where $x$ is a column vector of length $m$, $y$ is a row vector of ...
0answers
32 views

Derivative of an eigenvalue with respect to tensor itself

I have a rank-2 tensor $\mathbf{C}=C_{\alpha\beta} \mathbf{A}^\alpha \otimes \mathbf{A}^\beta$ defined in a curvilinear coordinate system. I want to compute the derivative of an eigenvalue $\Lambda_i$ ...
1answer
34 views

Why, given an object with rotational symmetry, is the axis of symmetry a principal axis?

When consulting textbooks and notes online about principle axes of inertia, I couldn't find a source which directly addressed the reasoning/proof behind following statement: "Given an object with a ...
1answer
40 views

Divergence theorem with tensors

I'm unsure how to apply the divergence theorem correctly on a volume integral of following type of object: $\partial_{a} \partial_{b} \partial_{c} T^{a b c}$. The volume integral is over a $n$-...
0answers
40 views

Generalizations of Positive Definiteness

What, if any, notions of positive definiteness can be extended to 3rd order tensors (and beyond)? The reason I ask is because the Hessian matrix of a convex function is positive semi-definite, but ...
1answer
35 views

Components of Maxwell tensor under Lorentz boost transformation

The following is taken from exercise 12.4 in D'Inverno. We wish to compute the transformation properties of the electric field and magnetic induction under a Lorentz boost. Given the following boost ...
1answer
24 views

Kronecker delta equation simplification

I am trying to simplify a tensor equation with Kronecker delta $$A_{ij} \big ( \delta_{ik}\delta_{jm} -\frac{1}{3}\delta_{ij}\delta_{km} \big)$$ $A$ and $\delta$ are Cartesian tensors. I know ...
0answers
17 views

Identity in continuum mechanics

For a problem in the textbook I am reading, I need to prove that $\int_Vw_{i,j}v_jdV = \int_Sw_iv_jn_jdS$, where $S$ is the boundary of the volume $V$, $v_i$ is the velocity vector field of a ...
2answers
59 views

Can we get a tensor out of summation of one vector with its transpose?

I don't know much about tensor calculus and here is something I'm trying to figure out. $$T=\mu({\nabla}\vec{V}+{\nabla}\vec{V}^T)$$ T is viscous stress tensor and $\vec{V}$ is the velocity vector. ...
0answers
17 views

Relation between torsion and curvature in Tensor calculus

In the context of tensor calculus by using Serret-Frenet formula or otherwise how to prove that $\tau^2=\displaystyle\frac{r'''^2}{k^2}-k^2-(\frac{k'}{k})^2$ where $\tau$ and $k$ represent ...
1answer
34 views

Vector norm in tensor notation

Just starting out with tensor notation. I'm trying to express a vector sum (norm) in tensor notation and its derivative. Suppose we have $s = \sum_k (a_k - b_k)^2$ What would be the natural way to ...
1answer
43 views

Tensor equation of a line

So, tensors appear to have no good description on the internet that actually starts from the basics. I've put together so far that, philosophically, tensors form a way to do your math in frames but ...
1answer
19 views

Describe the state of stress in a deformed elastic body with a given Cauchy Stress Tensor

How can I describe the state of stress in a deformed elastic body if the corresponding Cauchy stress tensor $T$ is constant and given by $T = −pI$, where $p > 0$ ??
1answer
31 views

An identity involving the derivative of a scalar function with respect to a tensor-valued variable [duplicate]

Let B be a tensor-valued variable, taking values from the set of second order tensors on the vector space naturally associated with Euclidean 3-space. It is given that B is invertible. I am looking ...
0answers
31 views

1answer
43 views

Number of Different Elements of $S_{ijkl}$ with Some Symmetries

I am not good at combinatorics so I am asking this simple question to learn a little. In fact, this question is motivated by the symmetries happening for the stiffness and Eshelby tensors in the ...
1answer
74 views

different approaches of defining tensors

This Wikipedia article says that tensor can be defined as miltilinear maps or be defined using tensor products. Could anybody explain with a simple example why these two approaches give the same ...
1answer
69 views

Why are contravariant vectors denoted with a superscript (and not a subscript)?

I wonder whether the choice of denoting contravariant/covariant vectors with a superscript/subscript is arbitrary (and could have been made the other way around), or whether there is a specific reason ...
0answers
30 views

1answer
45 views

Is this 2-tensor symmetric? It satisfies these conditions

I have some scalar field $p$ and a 2nd order tensor $\textbf{T}$ such that $div( \ \textbf{T} \ \textbf{grad}(p) \ )=0$ $\textbf{curl}(\ \textbf{T} \ \textbf{grad}(p) \ )=\textbf{0}$ $\textbf{T}$ ...
1answer
27 views

Is there an invariant similar to the characteristic polynomial for (0,2) and (2,0) tensors?

The characteristic polynomial of a matrix - a (1,1) tensor - is its invariant (independent on basis transformation). Is there a similar invariant for (0,2) and (2,0) tensors? The characteristic ...