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Definition/Summary
Covariant derivative, D, is a coordinate-dependent adjustment to ordinary derivative which makes each partial derivative of each coordinate unit vector zero: D\hat{\mathbf{e}}_i/\partial x_j\ =\ 0
The adjustment is made by a linear operator known both as the connection, \Gamma^i_{\ jk}, and as the Christoffel symbol, \{^{\ i\ }_{j\ k}\}.
Covariant derivative of the metric (g_{ij}) is zero.
Covariant derivative (unlike ordinary derivative) of any tensor is also a tensor (with an extra covariant index).
A vector V^i is parallel transported along a curve x^i(s) with tangent T^i(s) if its covariant directional derivative in the direction of that tangent is zero: (T.D)V^i/ds\ =\ 0
Equations
\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
\frac{D\,(a^i\hat{\mathbf{e}}_i)}{\partial x^j}\ =\ 0\ \text{for constant}\ a^i
\frac{Dg^{ij}}{\partial x^k}\ =\ 0
\Gamma^i_{\ jk}\ =\ \Gamma^i_{\ kj}
\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)
Parallel transport of a vector V^i along a curve x^i(s):
\frac{dx^j}{ds}\frac{DV^i}{\partial x^j}\ =\ 0\ \ \text{or}\ \ dV^i = -\Gamma^i_{\ jk}V^kdx^j
Geodesic deviation equation:
\frac{D^2\,\delta x^{\alpha}}{d\tau^2}\ =\ -\,R^{\alpha}_{\ \mu\beta\sigma}\,V^{\mu}\,V^{\sigma}\,\delta x^{\beta}
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, D, is a coordinate-dependent adjustment to ordinary derivative which makes each partial derivative of each coordinate unit vector zero: D\hat{\mathbf{e}}_i/\partial x_j\ =\ 0
The adjustment is made by a linear operator known both as the connection, \Gamma^i_{\ jk}, and as the Christoffel symbol, \{^{\ i\ }_{j\ k}\}.
Covariant derivative of the metric (g_{ij}) is zero.
Covariant derivative (unlike ordinary derivative) of any tensor is also a tensor (with an extra covariant index).
A vector V^i is parallel transported along a curve x^i(s) with tangent T^i(s) if its covariant directional derivative in the direction of that tangent is zero: (T.D)V^i/ds\ =\ 0
Equations
\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
\frac{D\,(a^i\hat{\mathbf{e}}_i)}{\partial x^j}\ =\ 0\ \text{for constant}\ a^i
\frac{Dg^{ij}}{\partial x^k}\ =\ 0
\Gamma^i_{\ jk}\ =\ \Gamma^i_{\ kj}
\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)
Parallel transport of a vector V^i along a curve x^i(s):
\frac{dx^j}{ds}\frac{DV^i}{\partial x^j}\ =\ 0\ \ \text{or}\ \ dV^i = -\Gamma^i_{\ jk}V^kdx^j
Geodesic deviation equation:
\frac{D^2\,\delta x^{\alpha}}{d\tau^2}\ =\ -\,R^{\alpha}_{\ \mu\beta\sigma}\,V^{\mu}\,V^{\sigma}\,\delta x^{\beta}
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!