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

Examples

See Also

Description

• 

Let M be a manifold, let  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  is T=EiTθi , where the vector fields E1,E2,...,En define a local frame on M with dual coframe θ1,θ2,...,θn. The tensor EiT is the directional covariant derivative of T with respect to  in the direction of Ei . The definition of the covariant derivative for sections of a vector bundle EM 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

withDifferentialGeometry:withTensor:

 

Example 1.

First create a 2 dimensional manifold M and define a connection C1 on the tangent space of M.

DGsetupx,y,M

frame name: M

(2.1)

C1ConnectionaD_x &t dx &t dybD_x &t dy &t dy+cD_y &t dy &t dx

C1:=aD_xdxdybD_xdydy+cD_ydydx

(2.2)

 

Define some tensor fields and compute their covariant derivatives with respect to C1.

T1evalDGy2D_x

T1:=y2D_x

(2.3)

CovariantDerivativeT1,C1

ay2+2yD_xdy

(2.4)

T2evalDGyxdx

T2:=yxdx

(2.5)

CovariantDerivativeT2,C1

ydxdxayxxdxdy+byxdydy

(2.6)

T3evalDGxdx &t D_y &t dx

T3:=xdxD_ydx

(2.7)

CovariantDerivativeT3,C1

bxdxD_xdxdy+cx+1dxD_ydxdx2axdxD_ydxdy+bxdxD_ydydy+bxdyD_ydxdy

(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.

XD_x

X:=D_x

(2.9)

DirectionalCovariantDerivativeX,T3,C1

cx+1dxD_ydx

(2.10)

ContractIndicesconvertX,DGtensor,CovariantDerivativeT3,C1,1,4

cx+1dxD_ydx

(2.11)

 

Example 2.

Define a frame on M and use this frame to specify a connection C2 on the tangent space of M.

FRFrameDatadxy,dyx,M1

FR:=dΘ1=xΘ1Θ2y,dΘ2=yΘ1Θ2x

(2.12)

DGsetupFR

frame name: M1

(2.13)

C2ConnectionE2 &t Θ1 &t Θ2

C2:=E2Θ1Θ2

(2.14)

 

Define some tensor fields and compute their covariant derivatives with respect to C2.

T4Θ2

T4:=Θ2

(2.15)

CovariantDerivativeT4,C2

Θ1Θ2

(2.16)

T5evalDGΘ2 &t E1 &t E2

T5:=Θ2E1E2

(2.17)

CovariantDerivativeT5,C2

Θ1E1E2Θ2+Θ2E2E2Θ2

(2.18)

 

Example 3.

First create a rank 3 vector bundle EM and define a connection C3 on E.

DGsetupx,y,u,v,w,E

frame name: E

(2.19)

C3ConnectionxD_v &t du &t dyyD_u &t dv &t dx

C3:=yD_udvdx+xD_vdudy

(2.20)

T6evalDGdu &t D_v

T6:=duD_v

(2.21)

CovariantDerivativeT6,C3

yduD_udx+ydvD_vdx

(2.22)

 

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

C4ConnectionD_x &t dy &t dx:

T7evalDGdx &t D_y &t du &t D_v:

CovariantDerivativeT7,C4,C3

dxD_xduD_vdxydxD_yduD_udx+ydxD_ydvD_vdxdyD_yduD_vdx

(2.23)

See Also

DifferentialGeometry, Tensor, Christoffel, Physics[Christoffel], Connection, ContractIndices, CurvatureTensor, Physics[Riemann], DirectionalCovariantDerivative


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