# 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 ...

69 views

### What are the non-vanishing directions of $A_{a(b} B_{cd)} - B_{a(b} A_{cd)}$? ($A$ and $B$ symmetric tensors)

Question Consider two symmetric tensors $A_{ab}=A_{ba},\, B_{ab}=B_{ba}$ which are generally of full rank (non-degenerate, all eigenvalues non-zero but of general sign) and their tensor product under ...
22 views

### Metric tensor for just one index

I'm novice in the territory of tensor calculus. I know the utilization of the metric tensor to transform the covariant basis to contravariant one, vice versa: $Z^i = Z^{ij}Z_j$ (Eq. 1) I am going ...
18 views

54 views

29 views

### Treatments of Lie theory and Noether theorems in tensor notation

I am looking for a conceptual treatment of Lie theory and Noether theorems that uses tensors calculus rather than exterior calculus. I know that tensor calculus is not optimal for these subjects, ...
59 views

20 views

### Index notation of tensors and mnemonics

I've been trying to learn to manipulate tensors but I've got probably too comfortable with all the matrices in my Linear Algebra course, that it gets really difficult beyond rank-3 tensors. So, ...
25 views

### Tensor products over Ring of Integers

I am having trouble computing the following tensor product(s): $$\mathcal {O}_{Q} \otimes _\mathbb {Z} \mathbb {C} =$$ Where in this case we use the ring of integers of the field of rationals, or ...
61 views

### Comodule induced from a comodule over a graded coalgebra

Let $G$ be a group with $1$ be the identity element of $G$. Let $(C,\bigtriangleup,\epsilon)$ be a coalgebra over a field $K$. $C$ is called $G$-graded if $C$ admits a decomposition as a direct sum of ...
48 views

In Spacetime & Geometry, Carrol immediately posits that, given that you have some tangent space vector (in the coordinate basis representation): $\frac{d}{d\lambda}=\frac{dx^{\mu}}{d\lambda}\... 1answer 48 views ### Is a tensor always identifiable as some symbol with some amount of indices? Say I have a (1,1) tensor$T^{\mu}_{\;\nu}$, I can represent it in some basis as follows (using Sean Carrol's notation):$T= T^{\mu}_{\;\nu} \; \hat{e}_{(\mu)} \otimes \hat{\theta}^{(\nu)}$If I let ... 1answer 34 views ### Interchange symmetry of a tensor of type$(0,4)$Suppose$A$is a tensor of type$(0,4)$on any manifold having the following symmetries $$A_{abcd}=-A_{bacd}=-A_{abdc}$$ and $$A_{abdc}+A_{acbd}+A_{adcb}=0.$$ How can I show that $$A_{abcd}=A_{... 0answers 44 views ### Christoffel symbols for the Poincaré ball model The metric tensor g_{ij} of the Poincaré ball model is$$ g_{ij} = \frac{\delta_{ij}}{(1 - x_k x^k)^2} $$where \delta_{ij} is the Kronecker delta and x^k are the ambient Cartesian coordinates.... 0answers 63 views ### Meaning of / intuition for contraction of tensors (in the Riemannian setting) I'm currently taking a course in differential geometry, and we are, I'm guessing, finally going to start working with the Riemannian curvature tensor after having covered a lot of smooth manifold ... 0answers 31 views ### Derivative of product tensor (or matrix product ) if I define a "product tensor" of two second-order tensors as the second-order tensor: \boldsymbol{C}=\boldsymbol{AB} such that C_{ij}=A_{ik}B_{kj} does the product rule applies to \dfrac{\... 0answers 13 views ### Changing the coordinates of a specific metric tensor As I have understood, the change of coordinates of a pseudo-Riemannian metric g(x) \to \bar g(\bar x) is given by:$$\tag{1} \bar g_{\alpha\beta}(\bar x) = \sum_{\mu,\nu} \dfrac{\partial x^\mu}{\... 1answer 73 views ### Curvature tensor for a particular Hilbert manifold My question involves an infinite dimensional Hilbert manifold with a Riemannian metric. My question is: What is the form of the curvature tensor for a infinite dimensional Hilbert manifold with ... 0answers 39 views ### Proving a vector identity using Einstein summation I need to prove $$\nabla \times [(u\cdot \nabla)u]=(u\cdot \nabla)(\nabla \times u)-[(\nabla \times u)\cdot \nabla]u+(\nabla\cdot u)(\nabla \times u)$$ So the ith component of each side using ... 1answer 48 views ### What is the definition of the tensor product$U⊗ V$of two vector spaces$U$and$V$? All I want to know is the exact definition of the vector space that is the tensor product of two vector spaces, say$U$and$V$, i.e. I want to know: - how its vectors are defined, - how does the$+$... 1answer 58 views ### Gradient of vector field notation Working in 3D. I know that the gradient is a vector operator defined as$\nabla = [\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}]$. The gradient of a scalar ... 1answer 31 views ### cardinality of orbits Let$G=\mbox{GL}(\mathbb{Z}/p\times\mathbb{Z}/p )\times \mbox{GL}(\mathbb{Z}/p\times\mathbb{Z}/p ) \times \mbox{GL}(\mathbb{Z}/p\times\mathbb{Z}/p )$,$p$prime, act on$(\mathbb{Z}/p\times \mathbb{Z}/...
Well, I've been learning Tensor algebra and related topics. So, I wonder tensor algebra is related to somehow familiar with me, so called group ring. $\mathbb{Z}$: a ring of integers. Let G be a ...
Let $\underline{u}$ be a $1$ - order tensor (say a column vector) I want to prove that : \$\underline{\operatorname{div}} \left( (\underline{\underline{\operatorname{grad}}} \, \underline{u})^T\...