calculate the covariant derivative of a tensor field in the direction of a vector field and with respect to a given connection - Maple Help

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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 EM

 

Description

Examples

See Also

Description

• 

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

 

XαY=XαYαXY.

 The definition of the directional covariant derivative operator X is extended to tensor fields on M as a derivation with respect to the tensor product.

• 

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

• 

Let EM be a vector bundle, let 1 be a linear connection on the tangent bundle of M and 2 be a connection on E. Let T be a mixed tensor on E, for example, T=Uτ, where U is a tensor field on M and 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 1 and 2 is XT=X1Uτ+UX2τ. 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

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)
M > 

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 vector fields and tensor fields and compute the directional covariant derivative with respect to C1.

M > 

X1D_y:

M > 

T1evalDGy2D_x

T1:=y2D_x

(2.3)
M > 

DirectionalCovariantDerivativeX1,T1,C1

ay2+2yD_x

(2.4)
M > 

X2D_y:

M > 

T2evalDGxD_y

T2:=xD_y

(2.5)
M > 

DirectionalCovariantDerivativeX2,T2,C1

bxD_x

(2.6)
M > 

X3D_y:

M > 

T3evalDGydx:

M > 

DirectionalCovariantDerivativeX3,T3,C1

ay1dx+bydy

(2.7)
M > 

X4evalDG2D_x3D_y

X4:=2D_x3D_y

(2.8)
M > 

T4evalDGydy &t dx:

M > 

DirectionalCovariantDerivativeX4,T4,C1

2cy+3ay3dydx3bydydy

(2.9)

 

Example 2.

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

 

M > 

FRFrameData1dxy,1dyx,M1:

M > 

DGsetupFR

frame name: M1

(2.10)
M1 > 

C2ConnectionE2 &t Θ1 &t Θ2

C2:=E2Θ1Θ2

(2.11)

 

Define a vector field and a tensor field and compute the directional covariant derivative with respect to C2.

M1 > 

X5evalDGx2E1y2E2:

M1 > 

T5evalDGE1 &t Θ2 &t E2

T5:=E1Θ2E2

(2.12)
M1 > 

DirectionalCovariantDerivativeX5,T5,C2

y2E1Θ1E2y2E2Θ2E2

(2.13)

 

Example 3.

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

M1 > 

DGsetupx,y,u,v,w,E

frame name: E

(2.14)
E > 

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

C3:=yD_udvdx+xD_vdudy

(2.15)
E > 

X6evalDGD_xD_y:

E > 

T6evalDGdu &t D_v

T6:=duD_v

(2.16)
E > 

DirectionalCovariantDerivativeX6,T6,C3

yduD_u+ydvD_v

(2.17)

 

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

E > 

C4ConnectionD_x &t dy &t dx

C4:=D_xdydx

(2.18)
E > 

X7evalDGD_x+2D_y

X7:=D_x+2D_y

(2.19)
E > 

T7evalDGdx &t D_y &t du &t D_v

T7:=dxD_yduD_v

(2.20)
E > 

DirectionalCovariantDerivativeX7,T7,C4,C3

dxD_xduD_vydxD_yduD_u+ydxD_ydvD_vdyD_yduD_v

(2.21)

See Also

DifferentialGeometry, Tensor, Christoffel, Connection, CovariantDerivative, CurvatureTensor, DGinfo, GeodesicEquations, ParallelTransportEquations


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