# 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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### Deriving the Epsilon-Tensor (Levi-Civita Symbol)

Given $$\hat{e}_i \times \hat{e}_j=\sum_{k=1}^3\epsilon_{ijk}\hat{e}_k$$ and $\hat{e}_1, \hat{e}_2, \hat{e}_3$ are the unit vectors in a right handed cartesian coordinate system ...
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### Show that $(2, 0)$ tensor is not a product of two vectors

The problem I am trying to solve is: Show that a general $(2, 0)$ tensor $K$, in $n$ dimensions, cannot be written as a direct product of two vectors, $A$ and $B$, but can be expressed as a sum of ...
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### 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 ...
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### Basic tensor derivation

Let $D$ be a tensor derivation on a mnaifold $M$. I have to show that if $D(\partial_i)=\sum F_i^j \partial_j$, then $D(dx^j)=-\sum F_i^j dx^i$. Any help on how to do this? Thanks in advance.
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### 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 ...
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### Levi-Civita and Matrix Determinant

I have the following relations: $$\epsilon_{lmn}\det{A} = \epsilon_{ijk}a_{li}a_{mj}a_{nk}$$ and $$\epsilon_{ijk}\det{A} = \epsilon_{lmn}a_{li}a_{mj}a_{nk}$$ I can see from the logic why these ...
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### Questions related dual vector and covector through a example.

I am finding dual vector very confusing to understand. I would like to understand it using example: Let $V=\Bbb{R}^3$, $v=\begin{bmatrix}2\\3\\5\\ \end{bmatrix} ∈ V$ and $f_1, f_2, f_3 ∈ V^*$ are ...