Articles of tensors

Simple question on symmetric tensors

This question seems to be silly, but i am really confused. Suppose we have a symmetric $2$-tensor $\omega$, I want to prove $$\omega(X,Y)=0\ \ \ \forall \ \ \ X,Y \iff \omega(X,X)=0 \ \ \forall \ \ \ \ X$$ The first direction is easy. The problem is the second direction i.e. $$\omega(X,X)=0 \ \ […]

Calculate the “inverse” of the Ricci tensor

Let $R_{\mu\nu}$ be the coordinates of the Ricci tensor. Using that $R^{\mu\alpha}R_{\alpha\nu}=\delta_\nu^\mu$ and $R_{\mu\nu}=R_{\nu\mu}$ it is possible to get the expressión of $R^{\mu\nu}$? Many thanks!

Connections and Ricci identity

Given $\nabla$ a torsionless connection, the Ricci identity for co-vectors is $$\nabla_a \nabla_b \lambda_c – \nabla_b \nabla_a \lambda_c = -R^d_{\,\,cab}\lambda_d.$$ Prove $R^a_{[bcd]} = 0$ by considering the co-vector field $\lambda_c = \nabla_c f$. By definition, $$R^a_{[bcd]} = 0 = \frac{1}{3!} \left(R^a_{\,\,bcd} + R^a_{\,\,cdb} + R^a_{\,\,dbc} – R^a_{\,\,bdc} – R^a_{\,\,cbd} – R^a_{\,\,dcb}\right)\,\,\,\,(1)$$ Attempt: Input the given […]

When is a symmetric 2-tensor field globally diagonalizable?

Suppose that $\mathbb{R}^n$ has a Riemannian metric $g$. Let $h$ be a smooth symmetric 2-tensor field on $\mathbb{R}^n$. At any point $p \in \mathbb{R}^n$, there is a basis of $T_p \mathbb{R}^n$ in which $h$ is diagonal. Is it always possible to find a global orthonormal frame $\{E_i\}$ that diagonalizes $h$? If not, what are the […]

Are Linear Transformations Always Second Order Tensors?

I’ve been reading a bit about tensors on Wikipedia (so correctness not guaranteed here) and I have a question. The order of a tensor $T$ is defined as $n+m$, where $n$ denotes the number of covariant indices and $m$ the number of contravariant indices. Wikipedia gives linear transformations as examples of second order tensors and […]

Difference between the Jacobian matrix and the metric tensor

I am just studying curvilinear coordinates and coordinate transformations. I have recently come across the metric tensor ($g_{ij}=\dfrac{\partial x}{\partial e_i}\dfrac{\partial x}{\partial e_j}+\dfrac{\partial y}{\partial e_i}\dfrac{\partial y}{\partial e_j}+\dfrac{\partial z}{\partial e_i}\dfrac{\partial z}{\partial e_j}$). As far as I understand it is used when transforming the arc element ds from one coordinate system e.g Cartesian to another one e.g. cylindrical […]

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. Our Lagrangian is $$L = g_{ab} {\dot{x}}^a {\dot{x}}^b = e^v \dot{t}^2 – e^{\lambda} \dot{r}^2 – r^2 \dot{\theta}^2 – r^2 \sin^2 \theta \dot{\phi}^2.$$ […]

Levi Cevita and Kronecker Delta identity

One of the popular Kronecker delta and Levi-Cevita identities reads $$\epsilon_{ijk}\epsilon_{ilm}=\delta_{jl}\delta_{km}-\delta_{kl}\delta_{jm}.$$ Now, is there an intuition or mnemonic that you use, that can help one learn these or similar mathematical identities more easily? Also, what is the motivation for expressing Levi-Cevita symbol in terms of Kronecker Delta in the first place?

Riemann tensor in terms of the metric tensor

The Riemann tensor is a function of the metric tensor when the Levi-Civita connection is used. Most tensor notation based texts give the Riemann tensor in terms of the Christoffel symbols, which are give in terms of the partial derivatives of the metric, but I have not seen the Riemann tensor given directly in terms […]

Understanding second derivatives

I am having a hard time understanding how to determine the second derivative of a matrix. I have researched Hessian matrices and do not see how i would apply it to vector funciton. problem statement: compute the derivative of the following $$ f(x) =\begin{bmatrix} x_1+x_1x_2^2\\ -x_2+x_2^2+x_1^2\\ \end{bmatrix}$$ I have found $$ DF =\begin{bmatrix} 1+x_2^2&2x_1\\ 2x_1&-1+x_2\\ […]