# Questions tagged [tensor-calculus]

Tensor calculus (tensor analysis) is a systematic extension of vector calculus to multivector and tensor fields in a form that is independent of the choice of coordinates on the relevant manifold, but which accounts for respective sub-spaces, their symmetries, and their connections.

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### Could Spacetime have a more General Geometry than just a Metric Field? [closed]

In standard general relativity the fundamental structure of spacetime is given through the six tuple $({M},\mathcal{O},\mathcal{A},{g},\nabla, {W}[\Phi,{g}])$, where ${M}$ is a smooth four-dimensional ...
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### Matrix index notation and Einstein summation

Can I ask why these two expressions are not equal? \begin{align}A_{ij}V^j&\ne V^kA_{ki}\\A_{ij}B^{ij}&\ne A^i{}_jB^j{}_i\end{align}
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### Transformation matrix from Cartesian to Polar of a Covariant vector

Suppose that there exists some contravariant 2-vector $V^{\alpha}$ . Now, let $V^{\alpha}$ be an element of the Cartesian coordinate system (x,y) such that it can be transformed into polar coordinates ...
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### What's exactly is moment of inertia?

I know that angular momentum can be expressed in terms of moment of inertia tensor as follows, $$\vec{L}= I_{\text{tensor}}\vec{w}$$ Where $I_{\text{tensor}}$ is tensor for moment of inertia. It can ...
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In the Stress-Charge form of Piezoelectric constitutive equations: $T$ $=$ $c$ : $S$ $-$ $(e^T) . E$ ( The symbols are explained in the picture below) My question is, why do we take the transpose of ... 2 votes 1 answer 47 views ### Why is viscous stress a tensorial quantity? In an incompressible fluid, the viscous stress (in Cartesian) is defined by \begin{align} \tau_{ij} = \eta(\partial_i v_j + \partial_j v_i) \end{align} for dynamic viscosity\eta$and velocity field$...
I was wondering if there were any ways of representing the metric tensor with a vector or scalar field and started calculating some potential ways. I recently stumbled across the equation \widetilde{...
### What do the symbols $\mu$ and $\nu$ mean in General Relativity?
I'm not an expert on general relativity, and below the tensors in the Einstein Field Equations, there are two confusing symbols: $\mu$ and $\nu$ below them. What do they mean? Any equations are ...