Tensor[CovariantDerivative] - calculate the covariant derivative of a tensor field with respect to a connection
CovariantDerivative(T, C1, C2)
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
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=∇EiT⊗θi , 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 E→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.
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 tensor fields and compute their covariant derivatives with respect to C1.
T1 ≔ evalDG⁡y2⁢D_x
T2 ≔ evalDG⁡y⁢x⁢dx
T3 ≔ evalDG⁡x⁢dx &t D_y &t dx
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 ≔ D_x
Define a frame on M and use this frame to specify a connection C2 on the tangent space of M.
FR ≔ FrameData⁡dxy,dyx,M1
frame name: M1
C2 ≔ Connection⁡E2 &t Θ1 &t Θ2
Define some tensor fields and compute their covariant derivatives with respect to C2.
T4 ≔ Θ2
T5 ≔ evalDG⁡Θ2 &t E1 &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
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:
T7 ≔ evalDG⁡dx &t D_y &t du &t D_v:
DifferentialGeometry, Tensor, Christoffel, Physics[Christoffel], Connection, ContractIndices, CurvatureTensor, Physics[Riemann], DirectionalCovariantDerivative
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