# 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.

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In rank 2, and given an inner product, the self-adjoint matrix is easy to define: $$\langle \mathbf{v}, A \mathbf{u} \rangle = \langle A \mathbf{v}, \mathbf{u} \rangle \tag{1}$$ If the inner product ...
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### Conceptual confusion regarding raising and lowering index

In Pavel Grinfeld's tensor calculus course (eg: 18:51) , it is said that in a given coordinate system on $\mathbb{R}^3$, we can write: $$e^i = g^{i \alpha} e_{\alpha}$$ Suppose we evaluate the above ...
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### Obtaining an explicit formula for a connection applied to an almost complex structure

Let $(M,J)$ be an almost complex manifold and $\nabla$ be a connection on $TM$. I am trying to see how we can obtain an explicit formula for $\nabla_X J$. I know that the way to extend $\nabla$ to a ...
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### The connection between a Metric Tensor and a Metric in Topology

Does the Metric tensor define a metric on the topology of the manifold/space it's on, and/or does the metric on the topology of the space define a metric tensor, if you go through all the trouble of ...
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### Tensor and Gauss divergence theorem

I am trying to see whether, in spherical coordinates, $$\int \left( \boldsymbol{r} \times \boldsymbol{\nabla} \cdot \boldsymbol{T} \right) \cdot \boldsymbol{e}_z dV$$ where $T$ is a 2D symmetric ...
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### Solving tensor version of homogeneous Sylvester equation

I have a system that reads (in summation convention) $$X_{j_1 j_2 ... j_N} R^{j_1}_{k_1} R^{j_2}_{k_2} ... R^{j_N}_{k_N} = X_{k_1 k_2 ... k_N}$$ where $N$ is a fixed value, $R$ a known matrix (...
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### Shape operator as a tensor field defined in a submanifold

Definition $4.18$ in O'Neill's book Semi-Riemannian geometry, with Applications to Relativity states that given a unit vector field $U$ normal to a hypersurface $M\subset \overline M$, then, there is ...
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### Is the covariant derivative a tensor?

While studying general relativity, the covariant derivative is constructed (in no rigorous manner) in order to make the derivative of a tensor transform like a tensor. Symbolically,  \nabla'_{\mu} V'...
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### A "trivial" question on tensors that I'm not sure about

(Notation $L^k$ is the space of $k$-covariant tensors) A friend posed this question to me - he's been studying about tensors and this question is an exercise there, claimed to be "trivial". ...
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### Tensors and the gradient - how to see that the gradient if formed by covariant elements and basis.

In Introduction to Tensor Analysis and the Calculus of Moving Surfaces by Pavel Grinfeld the gradient is addressed as a problem to be solved because both the coefficients and the basis vectors are ...
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### Any 2-tensor = Sum of a symmetric 2-tensor + an alternating 2-tensor

The following problem was asked in my assignment of linear algebra and I was not able to solve this. I am looking for hints here. Problem: Show that every 2-tensor can be uniquely written as sum of a ...
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### non-degenerate 2 forms, dimension and wedge product

Consider the problem from my assignment on tensor forms: Problem: We say that a 2-form n on a vector space V is non-degenerate of for $u\in V$ , n(u,v)=0 for all v$\in V$ implies u=0. Let w be a non-...
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### pullback map on k-tensors of a linear isomorphism and finding it's inverse

This question was asked in an assignment on k-tensor forms and I am having hard time solving this problem. I have been following my class notes which are based on the book of john lee. Question: Let V,...
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### How do I simplify $\delta_{ij} \delta^{jk}$?

How do I simplify $\delta_{ij} \delta^{jk}$? I know that $\delta_{ij} \delta_{jk}=\delta_{ik}$, but what do I do if the there's a Kronecker Delta symbol with upper indices and one with lower indices?
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### How can I find the tangent projection of a tensor field onto a surface orientation tensor?

Given a second order tensor that represents the orientation distribution of a set of vectors (specifically surface normals), $S_{ij} = \oint \hat n_i \hat n_j dA$, where $\hat n_i$ are surface ...
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