# 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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### Changing variables: partial derivatives of a tensor

Given is the tensor $T$ in Cartesian coordinates $T=\operatorname{diag}\{T_{xx},T_{yy},T_{zz}\}$ in cylindrical coordinates $T=\operatorname{diag}\{T_{rr},T_{\theta\theta},T_{zz}\}$ How does one ...
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### Show that $R_{\mu\nu}=fg_{\mu\nu}$ (Ricci and metric tensors) and $\dim(M)\geq 3$ then $f$ is constant

I need to prove that given the Ricci and metric tensors $R_{\mu\nu}=fg_{\mu\nu}$ and $\dim(M)\geq 3$ then $f$ is constant. I tried to use some identities but I end up with some sort of a proof ...
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### Cylindrical coordinate derivative of a vector field.

Considering the following identity transformation in cylindrical coordinate: $$\mathbf{v}(R,\theta,z)=R\;\mathbf{e}_{R}+\theta\;\mathbf{e}_{\theta}+z\;\mathbf{e}_{z}$$ Taking its derivative ...
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### Sectional curvature in a paraboloid is always positive.

I'm working on Lee's book ''Riemaniann Manifolds an Introduction to Curvature''. One exercise (11.1) is about to see that the paraboloid given by the equation $y=x_1^2+...+x_n^2$ has positive ...
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### Kronecker delta representation of a matrix (Quantum raising / lowering operators)

The Kronecker Delta is commonly used to represent a diagonal matrix: $$a_i \delta_{ij}=\left( \begin{array}{ccc} a_1 & 0 & 0\\ 0 & a_2 & 0\\ 0 & 0 & a_3 \end{array}\right)$$ ...
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### divergence theorem applied to a tensor dotted with a vector

Is my expression for the divergence theorem correct? $\int_{V}\underline{v}.div(\underline{\underline{\tau}})dV=\int_{S}\underline{v}.\left(\underline{\underline{\tau}}.\underline{n}\right)dS$ and ...
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### Does the Divergence Theorem hold for arbitrary tensor fields?

So, a heads up, this is my first post and I'm a fairly new user. Additionally, my math knowledge tops out at vector calculus and ODEs, but don't shy away from answering beyond my understanding should ...
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### Show $\delta_{KL}$ is a Cartesian tensor [closed]

By using the definition of kronecker delta $\delta_{KL}$, show that $\delta_{KL}$ is a Cartesian tensor, that is $$\delta ' _{MN}=L_{MK}L_{NL} \delta_{KL}$$ under the rotation $X_K=L_{MK}X' _M$. I ...
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### Derivative of dual basis vectors in terms of Christoffel symbols

How can I demonstrate from $$\frac{\partial \mathbf{e_j}}{\partial x^i} \equiv \Gamma_{ij}^k \mathbf{e_k}$$ what the value of $$\frac{\partial \mathbf{e^j}}{\partial x^i}$$ (with the index now ...
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### Show that $\epsilon_{ijk}\epsilon_{ljk}=2\delta_{il}$

Question: Show that $\epsilon_{ijk}\epsilon_{ljk}=2\delta_{il}$ where $\epsilon_{ijk}$ is the Levi-civita symbol and $\delta_{ij}$ is the Kronecker delta symbol. My attempt: $\epsilon_{ijk}$ assumes ...
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### Easy Tensor Notation: What is $\partial_\mu\partial^\nu x_\nu$

Part of a far larger piece of work... I know it should simplify $\partial^\nu x_\nu\partial_\mu$ but I don't know how. Similarly, does it change for the reverse? In other words I want to simplify: ...
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### How to Prove $(A \times B) \otimes C + (B \times C) \otimes A + (C \times A) \otimes B = [(A \times B) \cdot C] \bf{I}$?

Question Assume that $A$, $B$, $C$, and $D$ are four vectors in $\mathbb{R}^3$. Then I want to show that  {\bf{M}} \equiv (A \times B) \otimes C + (B \times C) \otimes A + (C \times A) \otimes ...
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### Programming nested sums in Matlab for graph-based statistic

I have an undirected graph $G=(E,N)$, where $E$ is the set of edges and $N$ is the set of nodes, of which $|N|=n$. It's convenient to represent the edges via a (symmetric) adjacency matrix $B$. I ...
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