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Tensor[DirectionalCovariantDerivative] - calculate the covariant derivative of a tensor field in the direction of a vector field and with respect to a given connection

Calling Sequences

DirectionalCovariantDerivative(X, T, C1, C2)

Parameters

X   - a vector field

T   - a tensor field

C1  - a connection

C2  - (optional) a second connection, needed when the tensor T is a mixed tensor defined on a vector bundle $E\to M$ Description

 • 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

$\left({\nabla }_{X}\mathrm{α}\right)\left(Y\right)=X\left(\mathrm{α}\left(Y\right)\right)-\mathrm{α}\left({\nabla }_{X}Y\right).$

The definition of the directional covariant derivative operator ${\nabla }_{X}$ is extended to tensor fields on $M$ as a derivation with respect to the tensor product.

 • 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 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 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. Examples

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

Example 1.

First create a 2 dimensional manifold $M$ and define a connection $\mathrm{C1}$ on the tangent space of $M$.

 > $\mathrm{DGsetup}\left(\left[x,y\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.1)
 M > $\mathrm{C1}≔\mathrm{Connection}\left(a\left(\mathrm{D_x}&t\mathrm{dx}\right)&t\mathrm{dy}-b\left(\mathrm{D_x}&t\mathrm{dy}\right)&t\mathrm{dy}+c\left(\mathrm{D_y}&t\mathrm{dy}\right)&t\mathrm{dx}\right)$
 ${\mathrm{C1}}{:=}{a}{}{\mathrm{D_x}}{}{\mathrm{dx}}{}{\mathrm{dy}}{-}{b}{}{\mathrm{D_x}}{}{\mathrm{dy}}{}{\mathrm{dy}}{+}{c}{}{\mathrm{D_y}}{}{\mathrm{dy}}{}{\mathrm{dx}}$ (2.2)

Define some vector fields and tensor fields and compute the directional covariant derivative with respect to $\mathrm{C1}$.

 M > $\mathrm{X1}≔\mathrm{D_y}:$
 M > $\mathrm{T1}≔\mathrm{evalDG}\left({y}^{2}\mathrm{D_x}\right)$
 ${\mathrm{T1}}{:=}{{y}}^{{2}}{}{\mathrm{D_x}}$ (2.3)
 M > $\mathrm{DirectionalCovariantDerivative}\left(\mathrm{X1},\mathrm{T1},\mathrm{C1}\right)$
 $\left({a}{}{{y}}^{{2}}{+}{2}{}{y}\right){}{\mathrm{D_x}}$ (2.4)
 M > $\mathrm{X2}≔\mathrm{D_y}:$
 M > $\mathrm{T2}≔\mathrm{evalDG}\left(x\mathrm{D_y}\right)$
 ${\mathrm{T2}}{:=}{x}{}{\mathrm{D_y}}$ (2.5)
 M > $\mathrm{DirectionalCovariantDerivative}\left(\mathrm{X2},\mathrm{T2},\mathrm{C1}\right)$
 ${-}{b}{}{x}{}{\mathrm{D_x}}$ (2.6)
 M > $\mathrm{X3}≔\mathrm{D_y}:$
 M > $\mathrm{T3}≔\mathrm{evalDG}\left(y\mathrm{dx}\right):$
 M > $\mathrm{DirectionalCovariantDerivative}\left(\mathrm{X3},\mathrm{T3},\mathrm{C1}\right)$
 ${-}\left({a}{}{y}{-}{1}\right){}{\mathrm{dx}}{+}{b}{}{y}{}{\mathrm{dy}}$ (2.7)
 M > $\mathrm{X4}≔\mathrm{evalDG}\left(2\mathrm{D_x}-3\mathrm{D_y}\right)$
 ${\mathrm{X4}}{:=}{2}{}{\mathrm{D_x}}{-}{3}{}{\mathrm{D_y}}$ (2.8)
 M > $\mathrm{T4}≔\mathrm{evalDG}\left(y\mathrm{dy}&t\mathrm{dx}\right):$
 M > $\mathrm{DirectionalCovariantDerivative}\left(\mathrm{X4},\mathrm{T4},\mathrm{C1}\right)$
 $\left({-}{2}{}{c}{}{y}{+}{3}{}{a}{}{y}{-}{3}\right){}{\mathrm{dy}}{}{\mathrm{dx}}{-}{3}{}{b}{}{y}{}{\mathrm{dy}}{}{\mathrm{dy}}$ (2.9)

Example 2.

Define a frame on $M$ and use this frame to specify a connection $\mathrm{C2}$ on the tangent space of $M$.

 M > $\mathrm{FR}≔\mathrm{FrameData}\left(\left[\frac{1\mathrm{dx}}{y},\frac{1\mathrm{dy}}{x}\right],\mathrm{M1}\right):$
 M > $\mathrm{DGsetup}\left(\mathrm{FR}\right)$
 ${\mathrm{frame name: M1}}$ (2.10)
 M1 > $\mathrm{C2}≔\mathrm{Connection}\left(\left(\mathrm{E2}&t\mathrm{Θ1}\right)&t\mathrm{Θ2}\right)$
 ${\mathrm{C2}}{:=}{\mathrm{E2}}{}{\mathrm{Θ1}}{}{\mathrm{Θ2}}$ (2.11)

Define a vector field and a tensor field and compute the directional covariant derivative with respect to $\mathrm{C2}$.

 M1 > $\mathrm{X5}≔\mathrm{evalDG}\left({x}^{2}\mathrm{E1}-{y}^{2}\mathrm{E2}\right):$
 M1 > $\mathrm{T5}≔\mathrm{evalDG}\left(\left(\mathrm{E1}&t\mathrm{Θ2}\right)&t\mathrm{E2}\right)$
 ${\mathrm{T5}}{:=}{\mathrm{E1}}{}{\mathrm{Θ2}}{}{\mathrm{E2}}$ (2.12)
 M1 > $\mathrm{DirectionalCovariantDerivative}\left(\mathrm{X5},\mathrm{T5},\mathrm{C2}\right)$
 ${{y}}^{{2}}{}{\mathrm{E1}}{}{\mathrm{Θ1}}{}{\mathrm{E2}}{-}{{y}}^{{2}}{}{\mathrm{E2}}{}{\mathrm{Θ2}}{}{\mathrm{E2}}$ (2.13)

Example 3.

First create a rank 3 vector bundle $E\to M$ and define a connection $\mathrm{C3}$ on $E$.

 M1 > $\mathrm{DGsetup}\left(\left[x,y\right],\left[u,v,w\right],E\right)$
 ${\mathrm{frame name: E}}$ (2.14)
 E > $\mathrm{C3}≔\mathrm{Connection}\left(x\left(\mathrm{D_v}&t\mathrm{du}\right)&t\mathrm{dy}-y\left(\mathrm{D_u}&t\mathrm{dv}\right)&t\mathrm{dx}\right)$
 ${\mathrm{C3}}{:=}{-}{y}{}{\mathrm{D_u}}{}{\mathrm{dv}}{}{\mathrm{dx}}{+}{x}{}{\mathrm{D_v}}{}{\mathrm{du}}{}{\mathrm{dy}}$ (2.15)
 E > $\mathrm{X6}≔\mathrm{evalDG}\left(\mathrm{D_x}-\mathrm{D_y}\right):$
 E > $\mathrm{T6}≔\mathrm{evalDG}\left(\mathrm{du}&t\mathrm{D_v}\right)$
 ${\mathrm{T6}}{:=}{\mathrm{du}}{}{\mathrm{D_v}}$ (2.16)
 E > $\mathrm{DirectionalCovariantDerivative}\left(\mathrm{X6},\mathrm{T6},\mathrm{C3}\right)$
 ${-}{y}{}{\mathrm{du}}{}{\mathrm{D_u}}{+}{y}{}{\mathrm{dv}}{}{\mathrm{D_v}}$ (2.17)

To covariantly differentiate a mixed tensor on $E$, a connection on $M$ is also needed.

 E > $\mathrm{C4}≔\mathrm{Connection}\left(\left(\mathrm{D_x}&t\mathrm{dy}\right)&t\mathrm{dx}\right)$
 ${\mathrm{C4}}{:=}{\mathrm{D_x}}{}{\mathrm{dy}}{}{\mathrm{dx}}$ (2.18)
 E > $\mathrm{X7}≔\mathrm{evalDG}\left(\mathrm{D_x}+2\mathrm{D_y}\right)$
 ${\mathrm{X7}}{:=}{\mathrm{D_x}}{+}{2}{}{\mathrm{D_y}}$ (2.19)
 E > $\mathrm{T7}≔\mathrm{evalDG}\left(\left(\left(\mathrm{dx}&t\mathrm{D_y}\right)&t\mathrm{du}\right)&t\mathrm{D_v}\right)$
 ${\mathrm{T7}}{:=}{\mathrm{dx}}{}{\mathrm{D_y}}{}{\mathrm{du}}{}{\mathrm{D_v}}$ (2.20)
 E > $\mathrm{DirectionalCovariantDerivative}\left(\mathrm{X7},\mathrm{T7},\mathrm{C4},\mathrm{C3}\right)$
 ${\mathrm{dx}}{}{\mathrm{D_x}}{}{\mathrm{du}}{}{\mathrm{D_v}}{-}{y}{}{\mathrm{dx}}{}{\mathrm{D_y}}{}{\mathrm{du}}{}{\mathrm{D_u}}{+}{y}{}{\mathrm{dx}}{}{\mathrm{D_y}}{}{\mathrm{dv}}{}{\mathrm{D_v}}{-}{\mathrm{dy}}{}{\mathrm{D_y}}{}{\mathrm{du}}{}{\mathrm{D_v}}$ (2.21) See Also