# Questions tagged [tensors]

For questions about tensors, tensor computation and specific tensors (e.g.curvature tensor, stress tensor). Tensor calculus is a technique that can be regarded as a follow-up on linear algebra. It is a generalisation of classical linear algebra. In classical linear algebra one deals with vectors and matrices.

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### Simplification of Levi-Civita Symbol

I need to simpify the following: $\begin{equation*} \varepsilon_{ijk}\dfrac{\partial^2 \phi}{\partial x_i \partial x_j} \end{equation*}$ For the Levi-Civita symbol $\varepsilon$. I tried expanding ...
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### Notational question on matrices, particularly in the context of differential geometry.

I've seen the notation both in notes, and in for example Loring W. Tu:s book, the notation $a^j_i$ for a matrix-element in a matrix $$A = \Big\{a^i_j\Big\}_{1 \leq i,j \leq n}$$ What is the row and ...
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### Trace of tensors in pseudo-Riemannian manifolds

I know that on a pseudo-Riemannian manifold $(M,g)$ a $(1,1)$ tensor field can be thought as an endomorphism of the tangent bundle. When one computes, for example, the trace of a $2$-covariant tensor, ...
1 vote
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### What formula can I use to translate $m$ indices into $n$ indices?

NOTE: The tensors are considered row-major. The sequence of elements is preserved. Let $A$ be a $m$-dimensional tensor of size $m_1 \times m_2 \times m_3 \cdots$ and $B$ be a $n$-dimensional tensor of ...
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### Why does the invariant 1-tensor integral gives 0 for any volume?

I was going through David Tong's vectors calculus notes. In Chapter 6.1.3 (Invariant Integrals) he gives the example Here are some examples. First, suppose that we have a 3d integral over the ...
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### in $ds^2 = g_{\mu \nu} dx^\mu dx^\nu$, why is $dx^\mu$ represented as a row vector and $dx^\nu$ represented as a column vector? [closed]

Just like what's in the title, in the equation $ds^2 = g_{\mu\nu} dx^\mu dx^\nu$, why are $dx^\mu$ and $dx^\nu$ represented differently?
### Coordinate expression for the codifferential of a $p$-form
I have been having difficulty obtaining the component expression of the codifferential of a $p$-form found on Wikipedia and in the book Riemannian Geometry and Geometric Analysis by Jürgen Jost. Let ...