# 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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### Does there exist a relationship between elasticity tensor $E_{ijkl}$ and $\int_{\Gamma} F^i z^i d \Gamma$?

Does there exist a relationship between elasticity tensor $$E_{ijkl}$$ and $$\int_{\Gamma} F^i z^i d \Gamma$$ where $F^i$ is force and $z^i$ displacement. $\Gamma$ is loaded boundary. I see these two ...
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### Meaning of Slot-Naming Index Notation (tensor conversion)

I'm studying the component representation of tensor algebra alone. There is a exercise question but I cannot solve it and cannot deduce answer from the text. The text is concise, I think it assumes a ...
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### What are covectors in special relativity?

In special relativity the purpose of vectors makes fairly intuitive sense, they represent a point in spacetime: $$x^{\mu}=\begin{pmatrix}x^0 \\ x^1 \\ x^2 \\ x^3\end{pmatrix}$$ and we can define the ...
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### Why do we write this tensor notation of space-time gradient contravariant tensor?

Why is $\partial^\mu=\frac{\partial}{\partial x_{\mu}}$ the contravariant component of space-time gradient four vector instad of $\partial^{\mu}=\frac{\partial}{\partial x^{\mu}}$?
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### Is the spacetime interval a tensor?

Tensors are objects that are invarient under a change of basis representation and whose coordinates change predictably. The spacetime interval is invarient under a change of coordinate representation ...
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### Definition of metric tensor and line elements

I know that the metric tensor $g_{ij}$ is defined as: $$g_{ij} = X_i \cdot X_j$$ where $x_i$ and $x_j$ are the covariant basis vectors ($X_i = \frac{\partial X}{\partial X^i}$) (the definition is ...
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### Why are isotropic tensors not considered scalars?

In introductory textbooks (Griffiths, Shankar, Boas) a tensor is introduced as a mathematical objects which transform in a specific manner under changes of basis (i.e. changes of the coordinate system)...
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### Tensor Method $SU(N)$

I'm working out the $SU(N)$ tensor method and reading Cheng-Li page 102, 103 (Sec. 4.3). I'm following the definition (4.94) which are $\psi^i=\psi_i^*$, $U_i^{.j}=U_{ij}$ and $U^i_{.j}=U_{ij}^*$ ...
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