 calculate the covariant derivative of a tensor field with respect to a connection - Maple Programming Help

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Tensor[CovariantDerivative] - calculate the covariant derivative of a tensor field with respect to a connection

Calling Sequences

CovariantDerivative(T, C1, C2)

Parameters

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

Description

 • 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 . 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.

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)
 > $\mathrm{C1}≔\mathrm{Connection}\left(a\left(\mathrm{D_x}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}-b\left(\mathrm{D_x}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}+c\left(\mathrm{D_y}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}\right)$
 ${\mathrm{C1}}{:=}{a}{}{\mathrm{D_x}}{}{\mathrm{dx}}{}{\mathrm{dy}}{-}{b}{}{\mathrm{D_x}}{}{\mathrm{dy}}{}{\mathrm{dy}}{+}{c}{}{\mathrm{D_y}}{}{\mathrm{dy}}{}{\mathrm{dx}}$ (2.2)

Define some tensor fields and compute their covariant derivatives with respect to $\mathrm{C1}$.

 > $\mathrm{T1}≔\mathrm{evalDG}\left({y}^{2}\mathrm{D_x}\right)$
 ${\mathrm{T1}}{:=}{{y}}^{{2}}{}{\mathrm{D_x}}$ (2.3)
 > $\mathrm{CovariantDerivative}\left(\mathrm{T1},\mathrm{C1}\right)$
 $\left({a}{}{{y}}^{{2}}{+}{2}{}{y}\right){}{\mathrm{D_x}}{}{\mathrm{dy}}$ (2.4)
 > $\mathrm{T2}≔\mathrm{evalDG}\left(yx\mathrm{dx}\right)$
 ${\mathrm{T2}}{:=}{y}{}{x}{}{\mathrm{dx}}$ (2.5)
 > $\mathrm{CovariantDerivative}\left(\mathrm{T2},\mathrm{C1}\right)$
 ${y}{}{\mathrm{dx}}{}{\mathrm{dx}}{-}\left({a}{}{y}{}{x}{-}{x}\right){}{\mathrm{dx}}{}{\mathrm{dy}}{+}{b}{}{y}{}{x}{}{\mathrm{dy}}{}{\mathrm{dy}}$ (2.6)
 > $\mathrm{T3}≔\mathrm{evalDG}\left(x\left(\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_y}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}\right)$
 ${\mathrm{T3}}{:=}{x}{}{\mathrm{dx}}{}{\mathrm{D_y}}{}{\mathrm{dx}}$ (2.7)
 > $\mathrm{CovariantDerivative}\left(\mathrm{T3},\mathrm{C1}\right)$
 ${-}{b}{}{x}{}{\mathrm{dx}}{}{\mathrm{D_x}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}\left({c}{}{x}{+}{1}\right){}{\mathrm{dx}}{}{\mathrm{D_y}}{}{\mathrm{dx}}{}{\mathrm{dx}}{-}{2}{}{a}{}{x}{}{\mathrm{dx}}{}{\mathrm{D_y}}{}{\mathrm{dx}}{}{\mathrm{dy}}{+}{b}{}{x}{}{\mathrm{dx}}{}{\mathrm{D_y}}{}{\mathrm{dy}}{}{\mathrm{dy}}{+}{b}{}{x}{}{\mathrm{dy}}{}{\mathrm{D_y}}{}{\mathrm{dx}}{}{\mathrm{dy}}$ (2.8)

To obtain a directional covariant derivative in the direction of a vector field $X$ from the covariant derivative, contract the last index of the covariant derivative against the vector field.

 > $X≔\mathrm{D_x}$
 ${X}{:=}{\mathrm{D_x}}$ (2.9)
 > $\mathrm{DirectionalCovariantDerivative}\left(X,\mathrm{T3},\mathrm{C1}\right)$
 $\left({c}{}{x}{+}{1}\right){}{\mathrm{dx}}{}{\mathrm{D_y}}{}{\mathrm{dx}}$ (2.10)
 > $\mathrm{ContractIndices}\left(\mathrm{convert}\left(X,\mathrm{DGtensor}\right),\mathrm{CovariantDerivative}\left(\mathrm{T3},\mathrm{C1}\right),\left[\left[1,4\right]\right]\right)$
 $\left({c}{}{x}{+}{1}\right){}{\mathrm{dx}}{}{\mathrm{D_y}}{}{\mathrm{dx}}$ (2.11)

Example 2.

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

 > $\mathrm{FR}≔\mathrm{FrameData}\left(\left[\frac{\mathrm{dx}}{y},\frac{\mathrm{dy}}{x}\right],\mathrm{M1}\right)$
 ${\mathrm{FR}}{:=}\left[{d}{}{\mathrm{Θ1}}{=}\frac{{x}{}{\mathrm{Θ1}}{}{\bigwedge }{}{\mathrm{Θ2}}}{{y}}{,}{d}{}{\mathrm{Θ2}}{=}{-}\frac{{y}{}{\mathrm{Θ1}}{}{\bigwedge }{}{\mathrm{Θ2}}}{{x}}\right]$ (2.12)
 > $\mathrm{DGsetup}\left(\mathrm{FR}\right)$
 ${\mathrm{frame name: M1}}$ (2.13)
 > $\mathrm{C2}≔\mathrm{Connection}\left(\left(\mathrm{E2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{Θ1}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{Θ2}\right)$
 ${\mathrm{C2}}{:=}{\mathrm{E2}}{}{\mathrm{Θ1}}{}{\mathrm{Θ2}}$ (2.14)

Define some tensor fields and compute their covariant derivatives with respect to $\mathrm{C2}$.

 > $\mathrm{T4}≔\mathrm{Θ2}$
 ${\mathrm{T4}}{:=}{\mathrm{Θ2}}$ (2.15)
 > $\mathrm{CovariantDerivative}\left(\mathrm{T4},\mathrm{C2}\right)$
 ${-}{\mathrm{Θ1}}{}{\mathrm{Θ2}}$ (2.16)
 > $\mathrm{T5}≔\mathrm{evalDG}\left(\left(\mathrm{Θ2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{E1}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{E2}\right)$
 ${\mathrm{T5}}{:=}{\mathrm{Θ2}}{}{\mathrm{E1}}{}{\mathrm{E2}}$ (2.17)
 > $\mathrm{CovariantDerivative}\left(\mathrm{T5},\mathrm{C2}\right)$
 ${-}{\mathrm{Θ1}}{}{\mathrm{E1}}{}{\mathrm{E2}}{}{\mathrm{Θ2}}{+}{\mathrm{Θ2}}{}{\mathrm{E2}}{}{\mathrm{E2}}{}{\mathrm{Θ2}}$ (2.18)

Example 3.

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

 > $\mathrm{DGsetup}\left(\left[x,y\right],\left[u,v,w\right],E\right)$
 ${\mathrm{frame name: E}}$ (2.19)
 > $\mathrm{C3}≔\mathrm{Connection}\left(x\left(\mathrm{D_v}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{du}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}-y\left(\mathrm{D_u}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dv}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}\right)$
 ${\mathrm{C3}}{:=}{-}{y}{}{\mathrm{D_u}}{}{\mathrm{dv}}{}{\mathrm{dx}}{+}{x}{}{\mathrm{D_v}}{}{\mathrm{du}}{}{\mathrm{dy}}$ (2.20)
 > $\mathrm{T6}≔\mathrm{evalDG}\left(\mathrm{du}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_v}\right)$
 ${\mathrm{T6}}{:=}{\mathrm{du}}{}{\mathrm{D_v}}$ (2.21)
 > $\mathrm{CovariantDerivative}\left(\mathrm{T6},\mathrm{C3}\right)$
 ${-}{y}{}{\mathrm{du}}{}{\mathrm{D_u}}{}{\mathrm{dx}}{+}{y}{}{\mathrm{dv}}{}{\mathrm{D_v}}{}{\mathrm{dx}}$ (2.22)

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

 > $\mathrm{C4}≔\mathrm{Connection}\left(\left(\mathrm{D_x}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dy}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx}\right):$
 > $\mathrm{T7}≔\mathrm{evalDG}\left(\left(\left(\mathrm{dx}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_y}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{du}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{D_v}\right):$
 > $\mathrm{CovariantDerivative}\left(\mathrm{T7},\mathrm{C4},\mathrm{C3}\right)$
 ${\mathrm{dx}}{}{\mathrm{D_x}}{}{\mathrm{du}}{}{\mathrm{D_v}}{}{\mathrm{dx}}{-}{y}{}{\mathrm{dx}}{}{\mathrm{D_y}}{}{\mathrm{du}}{}{\mathrm{D_u}}{}{\mathrm{dx}}{+}{y}{}{\mathrm{dx}}{}{\mathrm{D_y}}{}{\mathrm{dv}}{}{\mathrm{D_v}}{}{\mathrm{dx}}{-}{\mathrm{dy}}{}{\mathrm{D_y}}{}{\mathrm{du}}{}{\mathrm{D_v}}{}{\mathrm{dx}}$ (2.23)