# 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.

1,937 questions
Filter by
Sorted by
Tagged with
53 views

### Torsion tensor symmetrisation

Given that the affine connection can be written as: $$\Gamma ^a{}_{bc}= \bigg\{ {a \atop bc} \bigg\} - \frac{1}{2}(T^a{}_{bc}+T_c{}^a{}_b-T_{bc}{}^a)$$ Where $\big\{ {a \atop bc} \big\}$ denotes the ...
149 views

### If the world had four spatial dimensions, then area would be a tensor?

In three dimensions area is a vector because two dimensions have a direction relative to the third. If the world had four spatial dimensions then area would be a tensor? And what form then the laws of ...
76 views

### Wedge Product Convention

In Wald’s General Relativity textbook he defines the wedge product as: $$(w \wedge u)_{a_1 ... a_p b_1 ... b_q}= \frac{(p+q)!}{p!q!} w_{[a_1 ... a_p}u_{b_1 ... b_q]}$$ My question is relatively simple:...
68 views

### Definition of exterior derivative

In Sean Carroll's GR book, pg 84, the exterior derivative $d$ is defined as $$(dA)_{\mu_1\mu_2...\mu_{p+1}} = (p+1) \partial_{[\mu_1}A_{\mu_2...\mu_{p+1}]}\tag{2.76}$$ where $A$ is a $p$-form and the ...
58 views

51 views

### How to show ${\epsilon^{ij}}_k {\epsilon_{ij}}^l = -2\delta^l_k$ for the Levi-Civita symbol?
I am trying to prove the following identity for the levi civita symbol $${\epsilon^{ij}}_k {\epsilon_{ij}}^l = -2\delta^l_k,$$ taken from the Ashok Das QFT book pg 153, equation 4.102. I made use of ...