# 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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### 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, ...
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### Matrix inversion via Levi-Civita symbols

Using Cramer's formula for the inverse of a matrix $M_{ij}$, is it possible to express the entries $(M^{-1})_{ij}$ in terms of the entries $M_{ij}$ using the Levi-Civita symbol and Kronecker deltas? ...
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### what's the relationship of tensor and multivector

what's the relationship of multivector in geometric algebra and tensor? Is tensor a subset of multivector?
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### Solving systems of quadratic equations

There are efficient algorithms for solving a system of linear equations of the form $$\forall i \qquad 0 = a^i + \sum_j b^i_j x^j$$ or $$\mathbf{0} = \mathbf{a} + \mathbf{b} \cdot \mathbf{x}$$ Are ...
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### Understanding the definition of norm of tensors on a Riemannian manifold

I am teaching myself Riemannian Geometry in order to studying Mean Curvature flow. I was reading Lecture Notes on Mean Curvature Flow by Carlo Mantegazza and I'm trying understand the following ...
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### Are there any differences between tensors and multidimensional arrays?

I see lots of references saying things like A tensor is a multidimensional or N-way array But others that say things like it should be remarked that other mathematical entities occur in ...
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### What, Exactly, Is a Tensor?

I've repeatedly read things that reference tensors, and despite reading the wiki page and other answers here on stackexchange I still don't know what a tensor is. I'm fine with hearing things in the ...
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### 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 ...
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### How to intuitively understand parallel transport

In the article I've referenced below, and many other articles for that matter, the notion of parallel transport along a line of latitude $\theta=\theta_0$ on the unit 2-sphere is spoken about. What I ...
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### The Dimension of the Symmetric $k$-tensors

I want to compute the dimension of the symmetric $k$-tensors. I know that a covariant $k$-tensor $T$ is called symmetric if it is unchanged under permutation of arguments. Also, I know that the ...
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### Coordinate-free notation for tensor contraction?

I am not sure if I can prevent this question from being too vague or with too large an overlap with other similar math.SE questions, but I will do my best... A standard linear operation in tensor ...
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### Definition of a tensor for a manifold

While reading Nakahara's geometry, topology and physics. I came across the following definition of a tensor. A tensor $T$ of type $(p, q)$ is a multilinear map that maps $p$ dual vectors and $q$ ...
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### Are vectors and covectors the same thing?

In Euclidean space, we usually don't distinguish between vectors and covectors (or dual vectors or 1-forms or whatever you want to call them) -- because the spaces overlap. However, a physicist ...
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### Tensor Calculus

I am currently a 3rd year undergraduate electronic engineering student. I have completed a course in dynamics, calculus I, calculus II and calculus III. I've started self studying tensor calculus, my ...
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### Finding a basis for symmetric $k$-tensors on $V$

We say a function is $k$-linear if it takes $k$ values as input and is linear with respect to each of them. For example, determinant is a $n$-linear function. (If the matrix is $n \times n$) A tensor ...
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### Why is enveloping algebra called enveloping algebra?

What does the enveloping algebra of $\mathfrak{g}$ have to do with envelopes? If $\mathfrak{g}$ is a Lie algebra, we take tensor algebra on $\mathfrak{g}$ and make quotient through ideal of T, so we ...
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### Working out a concrete example of tensor product

From this entry in Wikipedia: The tensor product of two vector spaces $V$ and $W$ over a field $K$ is another vector space over $K$. It is denoted $V\otimes_K W$, or $V\otimes W$ when the ...
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