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Definition/Summary
Covariant derivative, [itex]D[/itex], is a coordinate-dependent adjustment to ordinary derivative which makes each partial derivative of each coordinate unit vector zero: [itex]D\hat{\mathbf{e}}_i/\partial x_j\ =\ 0[/itex]
The adjustment is made by a linear operator known both as the connection, [itex]\Gamma^i_{\ jk}[/itex], and as the Christoffel symbol, [itex]\{^{\ i\ }_{j\ k}\}[/itex].
Covariant derivative of the metric ([itex]g_{ij}[/itex]) is zero.
Covariant derivative (unlike ordinary derivative) of any tensor is also a tensor (with an extra covariant index).
A vector [itex]V^i[/itex] is parallel transported along a curve [itex]x^i(s)[/itex] with tangent [itex]T^i(s)[/itex] if its covariant directional derivative in the direction of that tangent is zero: [itex](T.D)V^i/ds\ =\ 0[/itex]
Equations
[tex]\frac{DV^i}{\partial x^j}\left(\text{also written}\ \frac{DV^i}{Dx^j}\ \text{or}\ \nabla_jV^i\ \text{or}\ V^i_{\ ;\,j}\right)\ =\ \frac{\partial V^i}{\partial x^j}\ -\ \Gamma^i_{\ jk}V^k[/tex]
[tex]\frac{D\,(a^i\hat{\mathbf{e}}_i)}{\partial x^j}\ =\ 0\ \text{for constant}\ a^i[/tex]
[tex]\frac{Dg^{ij}}{\partial x^k}\ =\ 0[/tex]
[tex]\Gamma^i_{\ jk}\ =\ \Gamma^i_{\ kj}[/tex]
[tex]\Gamma^i_{\ jk}\ =\ \{^{\ i\ }_{j\ k}\}\ =\ \frac{1}{2}\,g^{il}\left(\frac{\partial g_{jl}}{\partial x^k}\ +\ \frac{\partial g_{kl}}{\partial x^j}\ -\ \frac{\partial g_{jk}}{\partial x^l}\right)[/tex]
Parallel transport of a vector [itex]V^i[/itex] along a curve [itex]x^i(s)[/itex]:
[tex]\frac{dx^j}{ds}\frac{DV^i}{\partial x^j}\ =\ 0\ \ \text{or}\ \ dV^i = -\Gamma^i_{\ jk}V^kdx^j[/tex]
Geodesic deviation equation:
[tex]\frac{D^2\,\delta x^{\alpha}}{d\tau^2}\ =\ -\,R^{\alpha}_{\ \mu\beta\sigma}\,V^{\mu}\,V^{\sigma}\,\delta x^{\beta}[/tex]
Extended explanation
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
Covariant derivative, [itex]D[/itex], is a coordinate-dependent adjustment to ordinary derivative which makes each partial derivative of each coordinate unit vector zero: [itex]D\hat{\mathbf{e}}_i/\partial x_j\ =\ 0[/itex]
The adjustment is made by a linear operator known both as the connection, [itex]\Gamma^i_{\ jk}[/itex], and as the Christoffel symbol, [itex]\{^{\ i\ }_{j\ k}\}[/itex].
Covariant derivative of the metric ([itex]g_{ij}[/itex]) is zero.
Covariant derivative (unlike ordinary derivative) of any tensor is also a tensor (with an extra covariant index).
A vector [itex]V^i[/itex] is parallel transported along a curve [itex]x^i(s)[/itex] with tangent [itex]T^i(s)[/itex] if its covariant directional derivative in the direction of that tangent is zero: [itex](T.D)V^i/ds\ =\ 0[/itex]
Equations
[tex]\frac{DV^i}{\partial x^j}\left(\text{also written}\ \frac{DV^i}{Dx^j}\ \text{or}\ \nabla_jV^i\ \text{or}\ V^i_{\ ;\,j}\right)\ =\ \frac{\partial V^i}{\partial x^j}\ -\ \Gamma^i_{\ jk}V^k[/tex]
[tex]\frac{D\,(a^i\hat{\mathbf{e}}_i)}{\partial x^j}\ =\ 0\ \text{for constant}\ a^i[/tex]
[tex]\frac{Dg^{ij}}{\partial x^k}\ =\ 0[/tex]
[tex]\Gamma^i_{\ jk}\ =\ \Gamma^i_{\ kj}[/tex]
[tex]\Gamma^i_{\ jk}\ =\ \{^{\ i\ }_{j\ k}\}\ =\ \frac{1}{2}\,g^{il}\left(\frac{\partial g_{jl}}{\partial x^k}\ +\ \frac{\partial g_{kl}}{\partial x^j}\ -\ \frac{\partial g_{jk}}{\partial x^l}\right)[/tex]
Parallel transport of a vector [itex]V^i[/itex] along a curve [itex]x^i(s)[/itex]:
[tex]\frac{dx^j}{ds}\frac{DV^i}{\partial x^j}\ =\ 0\ \ \text{or}\ \ dV^i = -\Gamma^i_{\ jk}V^kdx^j[/tex]
Geodesic deviation equation:
[tex]\frac{D^2\,\delta x^{\alpha}}{d\tau^2}\ =\ -\,R^{\alpha}_{\ \mu\beta\sigma}\,V^{\mu}\,V^{\sigma}\,\delta x^{\beta}[/tex]
Extended explanation
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!