Let $M$ be a manifold and let $\nabla$ be a linear connection on the tangent bundle of $M$. If $X$ and $Y$ are vector fields on $M$, then ${\nabla }_{X}Y$ is a vector field on $M$ called the directional covariant derivative of $Y$ in the direction $X$ with respect to the connection $\nabla$. If $\mathrm{\α}$ is a differential 1-form, then ${\nabla }_X\mathrm{\α}$ is the 1-form defined by

Let $E\to M$ be a vector bundle and let $\nabla$ be a connection on $E$. If $X$ is a vector field on $M$ and $\mathrm{\σ}$is a section of $E$, then ${\nabla }_X\mathrm{\σ}$ is a section of $E$ called the directional covariant derivative of the section $\mathrm{\σ}$ in the direction $X$ with respect to the connection $\nabla$. The definition of the directional covariant derivative operator ${\nabla }_X$ is extended to tensor fields on the fibers of $E$ as above.

Let $E\to M$ be a vector bundle, let ${\nabla }^1$ be a linear connection on the tangent bundle of $M$ and ${\nabla }^2$ be a connection on $E$. Let $T$ be a mixed tensor on $E$, for example, $T\=U\otimes \mathrm{\τ}$, where $U$ is a tensor field on$M$ and $\mathrm{\τ}$ is a tensor field on the fibers of $E$. (In general $T$ will be a sum of such tensor products). Then the directional covariant derivative of $T$ in the direction $X$ with respect to the connections ${\nabla }^1$ and ${\nabla }^2$ is ${\nabla }_{X}T\={\nabla }_X^1\left(U\right)\otimes \mathrm{\τ}\+U\otimes {\nabla }_X^2\mathrm{\τ}$. This definition is extended to more general mixed tensors by linearity.

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

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

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

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

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

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

$\mathrm{X1}≔\mathrm{D\_y}\:$

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

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

$\mathrm{DirectionalCovariantDerivative}\left(\mathrm{X1}\,\mathrm{T1}\,\mathrm{C1}\right)$

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

$\mathrm{X2}≔\mathrm{D\_y}\:$

$\mathrm{T2}≔\mathrm{evalDG}\left(x\mathrm{D\_y}\right)$

$\mathrm{T2}:=x\mathrm{D\_y}$

$\mathrm{DirectionalCovariantDerivative}\left(\mathrm{X2}\,\mathrm{T2}\,\mathrm{C1}\right)$

$-bx\mathrm{D\_x}$

$\mathrm{X3}≔\mathrm{D\_y}\:$

$\mathrm{T3}≔\mathrm{evalDG}\left(y\mathrm{dx}\right)\:$

$\mathrm{DirectionalCovariantDerivative}\left(\mathrm{X3}\,\mathrm{T3}\,\mathrm{C1}\right)$

$-\left(ay-1\right)\mathrm{dx}\+by\mathrm{dy}$

$\mathrm{X4}≔\mathrm{evalDG}\left(2\mathrm{D\_x}-3\mathrm{D\_y}\right)$

$\mathrm{X4}:=2\mathrm{D\_x}-3\mathrm{D\_y}$

$\mathrm{T4}≔\mathrm{evalDG}\left(y\mathrm{dy}\&t\mathrm{dx}\right)\:$

$\mathrm{DirectionalCovariantDerivative}\left(\mathrm{X4}\,\mathrm{T4}\,\mathrm{C1}\right)$

$\left(-2cy\+3ay-3\right)\mathrm{dy}\mathrm{dx}-3by\mathrm{dy}\mathrm{dy}$

$\mathrm{FR}≔\mathrm{FrameData}\left(\left[\frac{1}{y}\mathrm{dx}\,\frac{1}{x}\mathrm{dy}\right]\,\mathrm{M1}\right)\:$

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

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

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

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

$\mathrm{X5}≔\mathrm{evalDG}\left(x^2\mathrm{E1}-y^2\mathrm{E2}\right)\:$

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

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

$\mathrm{DirectionalCovariantDerivative}\left(\mathrm{X5}\,\mathrm{T5}\,\mathrm{C2}\right)$

$y^2\mathrm{E1}\mathrm{\Θ1}\mathrm{E2}-y^2\mathrm{E2}\mathrm{\Θ2}\mathrm{E2}$

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

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

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

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

$\mathrm{X6}≔\mathrm{evalDG}\left(\mathrm{D\_x}-\mathrm{D\_y}\right)\:$

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

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

$\mathrm{DirectionalCovariantDerivative}\left(\mathrm{X6}\,\mathrm{T6}\,\mathrm{C3}\right)$

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

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

$\mathrm{C4}:=\mathrm{D\_x}\mathrm{dy}\mathrm{dx}$

$\mathrm{X7}≔\mathrm{evalDG}\left(\mathrm{D\_x}+2\mathrm{D\_y}\right)$

$\mathrm{X7}:=\mathrm{D\_x}\+2\mathrm{D\_y}$

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

$\mathrm{T7}:=\mathrm{dx}\mathrm{D\_y}\mathrm{du}\mathrm{D\_v}$

$\mathrm{DirectionalCovariantDerivative}\left(\mathrm{X7}\,\mathrm{T7}\,\mathrm{C4}\,\mathrm{C3}\right)$

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