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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3
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1answer
840 views

Invariant proof of the Contracted Bianchi Identity

In "Riemannian Manifolds: An Introduction to Curvature," John Lee states the following lemma: Lemma 7.7 (Contracted Bianchi Identity): The covariant derivatives of the Ricci and scalar curvatures ...
4
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2answers
1k views

Mathematically Precise Definition of Covariant and Contravariant Transformation

I am trying to understand the meanings of "covariant transformation" and "contravariant transformation" and how they are related. I have read the related Wikipedia article and still feel I cannot ...
5
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2answers
627 views

What is the definition of tensor contraction?

According to Wikipedia's page on tensor contraction: In general, a tensor of type $(m,n)$ (with $m \geq 1$ and $n \geq 1$) is an element of the vector space $V \otimes \ldots \otimes V \otimes V^* ...
1
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1answer
184 views

How did the author of the following paper compute the curvature matrix?

I would like to be shown how the curvature matrix K on page 7 in the paper "Regularisation Theory Applied to Neurofuzzy Modelling" (Bossley) is computed. ...
2
votes
1answer
44 views

Is there a particular name for a'long-small-small' tensor/array?

I'm thinking of a 3D array, with dimensions small,small,large. I've taken to saying 'sausage' as shorthand (and I'm sure there are worse NSFW descriptions) but is there a 'legitimate' description for ...
0
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2answers
340 views

Is correct to say that every tensor is a spinor but not every spinor is a tensor?

Can spinors be seen as a generalization of tensors,but with complex numbers?
1
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1answer
90 views

Matrix representing $\Lambda^k$(A)

Let V , W be finite dimensional vector spaces over R. Let A : V->W be a linear map. Choose bases of V and W and the corresponding bases of $\Lambda^k$(V ) and of $\Lambda^k$(W). How to show that the ...
1
vote
2answers
947 views

Prove the determinant of a tensor is invariant

Given is a second-order tensor $T$, and three arbitrary vectors, $u$, $v$ and $w$, defined in Euclidean point space $\mathcal{E}$. Prove that the determinant of the tensor $T$ $\det T=\frac{Tu.(Tv ...
0
votes
1answer
376 views

Relation between metric tensor and second fundamental form

I'm confused with these definitions. The metric of certain space and the second fundamental form seem to be the same object. I don't know what else to say, this is a pretty straight forward question. ...
1
vote
1answer
191 views

Trouble deriving the Harris Corner Detection

I just started studying a small paper about the Harris Corner Detection. The problem is I don't understand how step 7 is derived from step 6. In step 7 the expression is expanded in a way that we get ...
0
votes
1answer
221 views

gradient of row vector multiplied by scalar

I'm trying to re-write $v (u x)$ where $v$ and $u$ are row vectors and $x$ is a column vector as some expression $M x$ (or $\bar{v}x$, etc.). The motivation is because I'm trying to compute the ...
1
vote
1answer
229 views

Taylor expansion in time of the time component of a stress energy tensor

Perform a taylor expansion in 3 dimensions in time on the time compontent of of $T^{\alpha \beta}(t - r + n^{i} y_{i})$ given that $r$ is a contstant and $n^{i} y_{i}$ is the scalar product of a ...
2
votes
2answers
348 views

In tensor notation in Spivak's Calculus on Manifolds, what is that character that looks like a 3?

For example, saying that $T$ is a k-tensor one might see $T\in 3^k(V)$, of course it's not actually a 3. It looks somewhat like Fraktur font Z: $\frak{Z}$. I couldn't detexify it, and it doesn't ...
6
votes
2answers
2k views

Tensors as matrices vs. Tensors as multi-linear maps

So I read the answers in this question, and don't feel that much closer to an answer about how tensors as multi-linear maps and tensors as "multi-dimensional" matrices are truly related. For ...
45
votes
5answers
6k views

An Introduction to Tensors

As a physics student, I've come across mathematical objects called tensors in several different contexts. Perhaps confusingly, I've also been given both the mathematician's and physicist's definition, ...
10
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2answers
4k views

Intuitive way to understand covariance and contravariance in Tensor Algebra

I'm trying to understand basic tensor analysis. I understand the basic concept that the valency of the tensor determines how it is transformed, but I am having trouble visualizing the difference ...
1
vote
1answer
437 views

Einstein notation - difference between vectors and scalars

From Wikipedia: First, we can use Einstein notation in linear algebra to distinguish easily between vectors and covectors: upper indices are used to label components (coordinates) of ...
17
votes
1answer
852 views

Proof of $\det(\textbf{ST})=\det(\textbf{S})\det(\textbf{T})$ in Penrose graphical notation

For two matrices $\textbf{S}$ and $\textbf{T}$, a proof of $\det(\textbf{ST})=\det(\textbf{S})\det(\textbf{T})$ is given below in the diagrammatic tensor notation. Here $\det$ denotes the ...
21
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2answers
758 views

Is it misleading to think of rank-2 tensors as matrices?

Having picked up a rudimentary understanding of tensors from reading mechanics papers and Wikipedia, I tend to think of rank-2 tensors simply as square matrices (along with appropriate transformation ...
11
votes
5answers
3k views

How to visualize a rank-2 tensor?

The notion (rank-2) "tensor" appears in many different parts of physics, e.g. stress tensor, moment of inertia tensor, etc. I know mathematically a tensor can be represented by a 3x3 matrix. But I ...