# Questions tagged [tensors]

For questions about tensor, 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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### Derivative of exponential tensor with respect to a vector

Suppose you have a rank 4 Tensor $T$ whose coordinates are, in some basis, $T_{ijkl}$. Say the indices take values in some representation of $T$. Now suppose I construct a matrix by dotting this ...
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### How to evaluate these tensor operations?

I'm struggling to understand how to evaluate these expressions, despite being given their definitions inside the question. Can someone please help? EDIT: is the first one correct. I have just flipped ...
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### Confusion with tensor algebra in the context of special relativity and electrodynamics

I'm so confused with tensor algebra and I need help in finding the answer to part b. I honestly cannot figure out how to multiply tensors. Can someone please help/explain?
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### Induced mapping between tensor algebras

I'm dealing with tensor algebra in my differential geometry classes, and in particular with induced mappings. The mapping defined is the following: If $f:V\longrightarrow W$ is an isomorphism, then \...
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### Prove that $\bf RAR^T = A$ where $\bf A$ is a symmetric 2nd order tensor and ${\bf R} = {\bf I} - 2{\bf e} \otimes {\bf e}$

Here's how I've gotten so far $${\bf RAR^T} = {\bf A} - 2{\bf A} ({\bf e} \otimes {\bf e}) - 2 ({\bf e} \otimes {\bf e}) {\bf A} + 4({\bf Ae}\cdot{\bf e})({\bf e} \otimes {\bf e})$$ I can't quite ...
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### Tensor identity in orthogonal coordinate systems

In Riley, Hobson & Bence's "Mathematical methods for physics and engineering" third edition, in the chapter about tensors, one of the exercises involves finding the expression for the ...
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### Why do we need to use the chain rule in the transformation of a partial derivative of a vector component from different coordinate systems?

Let us consider the partial derivative of a vector component $A^i$, this is $\partial_jA^i = \frac{\partial A^i}{\partial X^j}$ We also know that we can write the transformed contravariant vector ...
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### What's is the best textbook to study tensor calculus for physicists?

I've finished studying tensor analysis from "Mathematical Methods in the physical sciences by: Mary L. Boas", but I wanna go furthermore in tensors that to be ready for General Relativity, ...
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### How to diagonalize a 2nd rank symmetric covariant tensor?

I've been self-teaching myself tensor mathematics by reading PDFs online, and I came across one that states that every 2nd rank symmetric covariant tensor can be diagonalized into a form where the ...
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### Tensor product in Paul r halmos book

This is the definition of tensor product in the book. But my question is "Does for every $z\in \scr U \otimes \scr V$ there exists $x\in \scr U$ , $y\in \scr V$ such that $z = x\otimes y ?$"...
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### Derivative of a matrix with a vector

Let there be three matrix F ,A (size (t+1) * t) and B (size t * t) and B is a function of x and x (size n) be a vector. $F=AB(x)$ Find the derivative of F w.r.t x. When I tried to find the ...
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### Ricci Curvature and Sectional Curvature

If we define the Ricci curvature for an orthonormal frame, we can simplify the sectional curvature formula to get: $$Ric(v, v) = \sum R(e_i, v)v \cdot e_i$$ However, I obtain a different formula to ...
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### Difference between Identity Matrix and Identity Tensor

I was thinking whether the identity matrix could be considered as a rank 2 tensor or not, but then I found out that there is a tensor called identity tensor that has the same function. So what do you ...
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### Linear Regression : functional multivariate or tensor

What would be the difference in treating data as tensor or as multivariate functional data? I have, for N days, the Energy Price Offer Curve for each hour of the day; I could see these data as N ...