Let $M$ be a manifold, let $\nabla$ be a linear connection on the tangent bundle of $M$, and let $T$ be a tensor field on $M$. The covariant derivative of $T$ with respect to $\nabla$ is $\nabla T\={\nabla }_{E_i}T\otimes {\mathrm{\θ}}^i$ , where the vector fields $E_1\,E_2\,..\.\,E_n$ define a local frame on $M$ with dual coframe ${\mathrm{\θ}}_1\,{\mathrm{\θ}}_2\,..\.\,{\mathrm{\θ}}_n$. The tensor ${\nabla }_{E_i}T$ is the directional covariant derivative of $T$ with respect to $\nabla$ in the direction of $E_{i }$. The definition of the covariant derivative for sections of a vector bundle $E\to M$ and for mixed tensors on $E$ is similar.

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form CovariantDerivative(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-CovariantDerivative.

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{Tensor}\right)\:$

$\mathrm{DGsetup}\left(\left[x\,y\right]\,M\right)$

$\mathrm{frame\; name:\; M}$

$\mathrm{C1}≔\mathrm{Connection}\left(a\left(\mathrm{D\_x}\&t\mathrm{dx}\right)\&t\mathrm{dy}-b\left(\mathrm{D\_x}\&t\mathrm{dy}\right)\&t\mathrm{dy}+c\left(\mathrm{D\_y}\&t\mathrm{dy}\right)\&t\mathrm{dx}\right)$

$\mathrm{C1}:=a\mathrm{D\_x}\mathrm{dx}\mathrm{dy}-b\mathrm{D\_x}\mathrm{dy}\mathrm{dy}\+c\mathrm{D\_y}\mathrm{dy}\mathrm{dx}$

$\mathrm{T1}≔\mathrm{evalDG}\left(y^2\mathrm{D\_x}\right)$

$\mathrm{T1}:=y^2\mathrm{D\_x}$

$\mathrm{CovariantDerivative}\left(\mathrm{T1}\,\mathrm{C1}\right)$

$\left(a{y}^2\+2y\right)\mathrm{D\_x}\mathrm{dy}$

$\mathrm{T2}≔\mathrm{evalDG}\left(yx\mathrm{dx}\right)$

$\mathrm{T2}:=yx\mathrm{dx}$

$\mathrm{CovariantDerivative}\left(\mathrm{T2}\,\mathrm{C1}\right)$

$y\mathrm{dx}\mathrm{dx}-\left(ayx-x\right)\mathrm{dx}\mathrm{dy}\+byx\mathrm{dy}\mathrm{dy}$

$\mathrm{T3}≔\mathrm{evalDG}\left(x\left(\mathrm{dx}\&t\mathrm{D\_y}\right)\&t\mathrm{dx}\right)$

$\mathrm{T3}:=x\mathrm{dx}\mathrm{D\_y}\mathrm{dx}$

$\mathrm{CovariantDerivative}\left(\mathrm{T3}\,\mathrm{C1}\right)$

$-bx\mathrm{dx}\mathrm{D\_x}\mathrm{dx}\mathrm{dy}\+\left(cx\+1\right)\mathrm{dx}\mathrm{D\_y}\mathrm{dx}\mathrm{dx}-2ax\mathrm{dx}\mathrm{D\_y}\mathrm{dx}\mathrm{dy}\+bx\mathrm{dx}\mathrm{D\_y}\mathrm{dy}\mathrm{dy}\+bx\mathrm{dy}\mathrm{D\_y}\mathrm{dx}\mathrm{dy}$

$X≔\mathrm{D\_x}$

$X:=\mathrm{D\_x}$

$\mathrm{DirectionalCovariantDerivative}\left(X\,\mathrm{T3}\,\mathrm{C1}\right)$

$\left(cx\+1\right)\mathrm{dx}\mathrm{D\_y}\mathrm{dx}$

$\mathrm{ContractIndices}\left(\mathrm{convert}\left(X\,\mathrm{DGtensor}\right)\,\mathrm{CovariantDerivative}\left(\mathrm{T3}\,\mathrm{C1}\right)\,\left[\left[1\,4\right]\right]\right)$

$\left(cx\+1\right)\mathrm{dx}\mathrm{D\_y}\mathrm{dx}$

$\mathrm{FR}≔\mathrm{FrameData}\left(\left[\frac{\mathrm{dx}}{y}\,\frac{\mathrm{dy}}{x}\right]\,\mathrm{M1}\right)$

$\mathrm{FR}:=\left[d\mathrm{\Θ1}\=\frac{x\mathrm{\Θ1}\bigwedge \mathrm{\Θ2}}{y}\,d\mathrm{\Θ2}\=-\frac{y\mathrm{\Θ1}\bigwedge \mathrm{\Θ2}}{x}\right]$

$\mathrm{DGsetup}\left(\mathrm{FR}\right)$

$\mathrm{frame\; name:\; M1}$

$\mathrm{C2}≔\mathrm{Connection}\left(\left(\mathrm{E2}\&t\mathrm{\Θ1}\right)\&t\mathrm{\Θ2}\right)$

$\mathrm{C2}:=\mathrm{E2}\mathrm{\Θ1}\mathrm{\Θ2}$

$\mathrm{T4}≔\mathrm{\Θ2}$

$\mathrm{T4}:=\mathrm{\Θ2}$

$\mathrm{CovariantDerivative}\left(\mathrm{T4}\,\mathrm{C2}\right)$

$-\mathrm{\Θ1}\mathrm{\Θ2}$

$\mathrm{T5}≔\mathrm{evalDG}\left(\left(\mathrm{\Θ2}\&t\mathrm{E1}\right)\&t\mathrm{E2}\right)$

$\mathrm{T5}:=\mathrm{\Θ2}\mathrm{E1}\mathrm{E2}$

$\mathrm{CovariantDerivative}\left(\mathrm{T5}\,\mathrm{C2}\right)$

$-\mathrm{\Θ1}\mathrm{E1}\mathrm{E2}\mathrm{\Θ2}\+\mathrm{\Θ2}\mathrm{E2}\mathrm{E2}\mathrm{\Θ2}$

$\mathrm{DGsetup}\left(\left[x\,y\right]\,\left[u\,v\,w\right]\,E\right)$

$\mathrm{frame\; name:\; E}$

$\mathrm{C3}≔\mathrm{Connection}\left(x\left(\mathrm{D\_v}\&t\mathrm{du}\right)\&t\mathrm{dy}-y\left(\mathrm{D\_u}\&t\mathrm{dv}\right)\&t\mathrm{dx}\right)$

$\mathrm{C3}:=-y\mathrm{D\_u}\mathrm{dv}\mathrm{dx}\+x\mathrm{D\_v}\mathrm{du}\mathrm{dy}$

$\mathrm{T6}≔\mathrm{evalDG}\left(\mathrm{du}\&t\mathrm{D\_v}\right)$

$\mathrm{T6}:=\mathrm{du}\mathrm{D\_v}$

$\mathrm{CovariantDerivative}\left(\mathrm{T6}\,\mathrm{C3}\right)$

$-y\mathrm{du}\mathrm{D\_u}\mathrm{dx}\+y\mathrm{dv}\mathrm{D\_v}\mathrm{dx}$

$\mathrm{C4}≔\mathrm{Connection}\left(\left(\mathrm{D\_x}\&t\mathrm{dy}\right)\&t\mathrm{dx}\right)\:$

$\mathrm{T7}≔\mathrm{evalDG}\left(\left(\left(\mathrm{dx}\&t\mathrm{D\_y}\right)\&t\mathrm{du}\right)\&t\mathrm{D\_v}\right)\:$

$\mathrm{CovariantDerivative}\left(\mathrm{T7}\,\mathrm{C4}\,\mathrm{C3}\right)$

$\mathrm{dx}\mathrm{D\_x}\mathrm{du}\mathrm{D\_v}\mathrm{dx}-y\mathrm{dx}\mathrm{D\_y}\mathrm{du}\mathrm{D\_u}\mathrm{dx}\+y\mathrm{dx}\mathrm{D\_y}\mathrm{dv}\mathrm{D\_v}\mathrm{dx}-\mathrm{dy}\mathrm{D\_y}\mathrm{du}\mathrm{D\_v}\mathrm{dx}$