# A tensor crib sheet

Posted by on Jan 23, 2013 in Science | 0 comments

Here are a few useful formulae for work with tensors:

$\overset{\rightharpoonup }{e}^i\cdot \overset{\rightharpoonup }{e}_j=\delta _j^i$

$\overset{\rightharpoonup }{e}_i\cdot \overset{\rightharpoonup }{e}^j=\delta _i^j$

$\overset{\rightharpoonup }{e}_i\cdot \overset{\rightharpoonup }{e}_j=g_{\text{ij}}$

$\overset{\rightharpoonup }{e}^i\cdot \overset{\rightharpoonup }{e}^j=g^{\text{ij}}$

$\overset{\rightharpoonup }{V}=v^i\overset{\rightharpoonup }{e}_i=v_i\overset{\rightharpoonup }{e}^i$

$\delta _n^m=g^{\text{mk}}g_{\text{kn}}$

$\frac{\partial \overset{\rightharpoonup }{e}_m}{\partial y^n}=\Gamma _{\text{mn}}^k\overset{\rightharpoonup }{e}_k$

$\frac{\partial \overset{\rightharpoonup }{e}_m}{\partial y^n}\overset{\rightharpoonup }{e}_k=\Gamma _{\text{mn}}^k\overset{\rightharpoonup }{e}_k$

$\overset{\rightharpoonup }{e}_m=\frac{\partial \overset{\rightharpoonup }{r}}{\partial y^m}$

$\overset{\rightharpoonup }{e} ^m= \overset{\rightharpoonup }{\nabla }y^m$

$\Gamma _{\text{ij}}^m=\frac{1}{2}g^{\text{mk}}\left[\frac{\partial g_{\text{ki}}}{\partial y^j}+\frac{\partial g_{\text{jk}}}{\partial y^i}-\frac{\partial g_{\text{ij}}}{\partial y^k}\right]$

$g_{nm} = g_{mn}$

$g^{nm} = g^{mn}$

$\Gamma _{\text{ij}}^m = \Gamma _{\text{ji}}^m$

${\Gamma '}_{ij}^k=\frac{\partial {y'}^k}{\partial y^t}\frac{\partial y^r}{\partial {y'}^i}\frac{\partial y^s}{\partial {y'}^j} \Gamma_{rs}^t+\frac{\partial {y'}^k}{\partial y^r}\frac{\partial^2 y^r}{\partial {y'}^i\partial {y'}^j}$

aka the Christoffel symbol is not a tensor

$\nabla _j\overset{\rightharpoonup }{V}=\left[\frac{\partial v^i}{\partial y^j}+\Gamma _{\text{kj}}^iv^k\right]\overset{\rightharpoonup }{e}_i$

$\nabla _j\overset{\rightharpoonup }{V}=\left[\frac{\partial v_i}{\partial y^j}-\Gamma _{\text{ij}}^kv^k\right]\overset{\rightharpoonup }{e}^i$