Tensor[DirectionalCovariantDerivative] - calculate the covariant derivative of a tensor field in the direction of a vector field and with respect to a given connection
DirectionalCovariantDerivative(X, T, C1, C2)
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→M
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
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 E→M 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 E→M 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=U⊗τ, 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=∇X1U⊗τ+U⊗∇X2τ. 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.
First create a 2 dimensional manifold M and define a connection C1 on the tangent space of M.
frame name: M
C1 ≔ Connection⁡a⁢D_x &t dx &t dy−b⁢D_x &t dy &t dy+c⁢D_y &t dy &t dx
Define some vector fields and tensor fields and compute the directional covariant derivative with respect to C1.
X1 ≔ D_y:
T1 ≔ evalDG⁡y2⁢D_x
X2 ≔ D_y:
T2 ≔ evalDG⁡x⁢D_y
X3 ≔ D_y:
T3 ≔ evalDG⁡y⁢dx:
X4 ≔ evalDG⁡2⁢D_x−3⁢D_y
T4 ≔ evalDG⁡y⁢dy &t dx:
Define a frame on M and use this frame to specify a connection C2 on the tangent space of M.
FR ≔ FrameData⁡1⁢dxy,1⁢dyx,M1:
frame name: M1
C2 ≔ Connection⁡E2 &t Θ1 &t Θ2
Define a vector field and a tensor field and compute the directional covariant derivative with respect to C2.
X5 ≔ evalDG⁡x2⁢E1−y2⁢E2:
T5 ≔ evalDG⁡E1 &t Θ2 &t E2
First create a rank 3 vector bundle E→M and define a connection C3 on E.
frame name: E
C3 ≔ Connection⁡x⁢D_v &t du &t dy−y⁢D_u &t dv &t dx
X6 ≔ evalDG⁡D_x−D_y:
T6 ≔ evalDG⁡du &t D_v
To covariantly differentiate a mixed tensor on E, a connection on M is also needed.
C4 ≔ Connection⁡D_x &t dy &t dx
X7 ≔ evalDG⁡D_x+2⁢D_y
T7 ≔ evalDG⁡dx &t D_y &t du &t D_v
DifferentialGeometry, Tensor, Christoffel, Connection, CovariantDerivative, CurvatureTensor, DGinfo,
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