# 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 ...

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### addition of tensors

The tensor algebra over a $n$-dimensional vector space $E$ is defined as the direct sum: $$\bigotimes E:=\bigoplus_{r,s>0}\Big(\bigotimes\nolimits_r^sE\Big)$$ An element of $\bigotimes E$ is ...
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### What does shear mean?

As I understand it, the gradient of a vector field can be decomposed into parts that relate to the divergence, curl, and shear of the function. I understand what divergence and curl are (both ...
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### surface gradient of the unit normal vector

I am reading a book that defines a curvature tensor as $\boldsymbol{K} = -\nabla_\pi \boldsymbol{\hat{n}}$ where $\boldsymbol{\hat{n}}$ is the unit normal vector of a surface and \begin{align*} \...
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### Does this calculus argument involving rank-4 tensors make sense?

Edit: Completely rewritten to be shorter and easier to digest. Background and the Actual Question: I'm trying to derive a gradient formula (back propagation) for a machine learning application. The ...
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### Intuition about antisymmetrizing tensor equations

I was looking at the symmetries of the Riemann tensor, and tried to prove a couple of properties, namely If $\nabla$ is torsion-free, then: (i) $R^a_{\,[bcd]}=0$, and (ii) $R^a_{\,b[cd;e]}=0$. ...
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### Let $V$ and $W$ be vector spaces over $\mathbb{C}$. Show that $0\otimes w = v\otimes 0 = 0 \in V \otimes W$.

Algebraically, the vector space $V \otimes W$ is spanned by elements of the form $v \otimes w$, and the following rules are satisfied, for any scalar $c$. The definition is the same no matter which ...
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### How to represent matrix multiplication in tensor algebra?

How can we represent matrix multiplication in tensor algebra? Even if we assume all matrices represent contravariant tensors only, clearly matrix multiplication does not correspond to the ...
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### What is the “taxonomy” or “hierarchy” (partial ordering) of algebraic objects used to attempt to capture geometric intuition? [closed]

What follows is a list of terms all of whose relationships to one another I have never fully succeeded in establishing, despite having spent much of 6-8 years trying to so. Feel no need to give ...