# 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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### Vector valued 2-forms which satisfy Jacobi Identity

Motivated by this MO question we ask the following two questions: 1)What is an example of a compact manifold $M$ which does not admit any smooth (1,2) tensor $\omega$ which restriction to each ...
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### Geometric Significance of some features of the Exterior Algebra

I've been tinkering with differential forms for a while now, and I've had a few questions all rolled into one trying to understand them. The exterior derivative is quite natural to me - it looks just ...
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### Eigendecomposition and eigenvalue scaling for tensor networks

I've recently been interested in studying tensors and networks of tensors. Consider the following quantity: $$Tr(A^N)$$ Where $A$ is a square matrix and $N$ is large. This is extremely easy - all we ...
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### Einstein notation

I'm confused about a specific issue that I have with the Einstein notation (for tensor fields on manifolds). I want to write the following thing: Let $X$ be a smooth manifold. Choosing local ...
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### Matrix Calculus

My school offers Matrix Calculus class next semester. I have never heard about this subject before and got intrigued. After a short chat with professor I found myself unable to get rid of suspicion ...
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### Is there a method of knowing which of the Christoffel symbols of second kind survive and vanish for a given metric?

I'm trying to solve a problem and the given $g_{ij}$ metric is $$\left[\begin{array}{cc}1&0&0\\0&(x^1)^2&0\\0&0&(x^1 \sin x^2)^2\end{array}\right]$$ The non-zero Christoffel ...
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### Intuition about the Riemann and Ricci tensors

I have started an attempt to self-study Riemannian Geometry. I well understand all the algebraic properties of the Riemann tensor (With a symmetric connection), and how it gradually becomes less and ...
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### What is tensor, really?

How can one understands the definition of tensor from the purely formal point of view? To what abstract structure this concept can be generalized?
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### Multilinear or Tensor Regression?

Given input data $x_t\in \mathbb{R}^n$ and output data $y_t\in\mathbb{R}^m$, the closed form solution to $\min_A \sum_t \|y_t - Ax_t\|^2_2$ is given by $A = (XX^T)^{-1}XY^T$ where $x_t$ form the ...
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### Use of graph theory to determine tensor contraction ordering

I am considering using a computer program to execute tensor contractions like the following: $\displaystyle \sum_{ij}^{o} \sum_{ab}^{v} \sum_{KL}^{X} B_{ia}^{K} B_{ia}^{L} B_{jb}^{K} B_{jb}^{L}$ ...
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### Proving a vector identity using Einstein summation

I need to prove $$\nabla \times [(u\cdot \nabla)u]=(u\cdot \nabla)(\nabla \times u)-[(\nabla \times u)\cdot \nabla]u+(\nabla\cdot u)(\nabla \times u)$$ So the ith component of each side using ...
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### Equivalent definitions of tensors on finite dimensional spaces.

I have recently been studying various texts on differential geometry, and I am quite puzzled that various authors define the notion of a tensor quite differently. I have come across the following ...
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### How do I compute the gradient of a tensor?

From this paper, we have three matrices $U\in \mathbb{R}^{n\times d_U}$, $M\in \mathbb{R}^{m\times d_m}$, $C\in \mathbb{R}^{c\times d_C}$ and a tensor $S\in \mathbb{R}^{d_U \times d_M \times d_C}$, ...
Menzel defines covariant differentiation as equivalent to partial differentiation with respect to the general coordinates. “To indicate the covariant nature of the differential operator, set $$\frac{\... 0answers 28 views ### Tensor Eigenstates I have the following equation:$$f_i^´(t+1)=\sum_{jk}R_{i|jk}\tilde{f_j}(t)\tilde{f_k}(t)$$It is about evolution of a population. I use this equation in my python program in the following way: <... 0answers 23 views ### Raising index on covariant derivative So suppose X is some vector field and t is a tangent vector to some curve on some smooth manifold. Then t^a\nabla_a X gives the directional derivative of the vector field in the direction of t.... 0answers 40 views ### Derivative of an eigenvalue with respect to tensor itself I have a rank-2 tensor \mathbf{C}=C_{\alpha\beta} \mathbf{A}^\alpha \otimes \mathbf{A}^\beta defined in a curvilinear coordinate system. I want to compute the derivative of an eigenvalue \Lambda_i ... 0answers 69 views ### What are the non-vanishing directions of A_{a(b} B_{cd)} - B_{a(b} A_{cd)}? (A and B symmetric tensors) Question Consider two symmetric tensors A_{ab}=A_{ba},\, B_{ab}=B_{ba} which are generally of full rank (non-degenerate, all eigenvalues non-zero but of general sign) and their tensor product under ... 0answers 48 views ### Why is the transformation law for tangent space coordinate basis vectors “easily seen” using the chain rule? In Spacetime & Geometry, Carrol immediately posits that, given that you have some tangent space vector (in the coordinate basis representation): \frac{d}{d\lambda}=\frac{dx^{\mu}}{d\lambda}\... 0answers 33 views ### Tensor formula in SU(3) representations I am trying to understand the Georgi chapter of tensor methods in SU(3) representations, and I don't know how to resolve the tensor product of 2 matrices in a 2 heavy quark + 2 light antiquark ... 0answers 125 views ### Are quaternions used in tensor analysis? At the moment I am learning tensor analysis. In the books I study (civil engineering/mechanical books on tensors) there is a lot written about rotation matrices/tensors, however quaternions are not ... 0answers 56 views ### Connecting physical tensors to mathematical tensors I feel like I (maybe) understand the mathematical /algebraic perspective on tensors, for example as described in Wikipedia/Monoidal Category. You need a (simultaneously left & right) unit and an ... 0answers 45 views ### Tensor varieties? I read somewhere that the space of rank-one tensors, known as the Segre variety, defined by$$ Seg: \mathbb{P}V_1 \times \cdots \times \mathbb{P}V_n \rightarrow \mathbb{P}(V_1 \otimes \cdots \otimes ...
Integration is only possible for forms and not for general tensors? What is the true reason for this? Or can integration of $k$-forms be extended in some natural way to arbitrary $(k,l)$ tensors;if so ...