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I. Metric and Connection
On scalar functions the coderivative ${D_\mu } =
{{\mathbf{g}}_\mu } \cdot D$, $D = {{\mathbf{g}}^\mu }{D_\mu
}$, acts as an ordinary spacetime derivative ${D_\mu
}f({\mathbf{x}}): = {\partial _\mu }f({\mathbf{x}})$ . With this
rule and the Levi-Civita connection as defined in (1.14), covariant
derivatives of a vector ${\mathbf{A}}$ are calculated as:
\[ {D_\mu }{\mathbf{A}} =
{D_\mu }({A^\nu } {{\mathbf{g}}_\nu }) = ({\partial _\mu }{A^\nu } +
\Gamma _{\mu \kappa }^\nu {A^\kappa }) {{\mathbf{g}}_\nu }: = {A^\nu
}_{;\mu }{{\mathbf{g}}_\nu } \]
\[ D \cdot {\mathbf{A}}
= {{\mathbf{g}}^\mu } {D_\mu } \cdot ({A^\nu }{{\mathbf{g}}_\nu }) =
{\partial _\mu }{A^\mu } + \Gamma _{\mu \kappa }^\mu {A^\kappa
} \]
\[ {\mathbf{a}} \cdot
D{\mathbf{A}} = {a^\mu }{D_\mu }({A^\nu }{{\mathbf{g}}_\nu }) =
{a^\mu }({\partial _\mu }{A^\nu } + \Gamma _{\mu \kappa }^\nu
{A^\kappa }){{\mathbf{g}}_\nu } \]
\[ D \wedge {\mathbf{A}}
= {{\mathbf{g}}^\mu }\wedge {D_\mu } ({A_\nu }{{\mathbf{g}}^\nu })=
{\partial _\mu }{A_\nu } {{\mathbf{g}}^\mu } \wedge
{{\mathbf{g}}^\nu } + {A_\nu }D \wedge {{\mathbf{g}}^\nu } \]
The last term of (1.20) is zero in GRT; see equation (1.26).
If and only if that is the case, the covariant curl $D \wedge $ is
equivalent to the exterior derivative ${\text{d}}$ in (1.7). Note
the standard notation ‘;’ in (1.17) for the covariant derivative in
the component formalism.
If ${\mathbf{F}}$ is the 2-form (1.8), then:
\[ \begin{gathered} D
\cdot {\mathbf{F}} = {{\mathbf{g}}^\mu } \cdot \left[ {{D_\mu
}\left( {\frac{1}{2} {F^{\kappa \lambda }}{{\mathbf{g}}_\kappa }
\wedge {{\mathbf{g}}_\lambda }} \right)} \right] \\ \qquad \qquad
\qquad = \left( {{\partial _\kappa }{F^{\kappa \lambda }} + \Gamma
_{\nu \kappa }^\nu {F^{\kappa \lambda }}}
\right){{\mathbf{g}}_\lambda } {\text{:}} = {F^{\kappa \lambda
}}_{;\kappa }{{\mathbf{g}}_\lambda } \end{gathered} \]
\[ D \cdot D \cdot
{\mathbf{F}} = {g^\mu } \cdot {D_\mu }({F^{\kappa \lambda
}}_{;\kappa }{{\mathbf{g}}_\lambda }) = {F^{\mu \nu }}{\partial _\nu
}\Gamma _{\kappa \mu }^\kappa \]
The last term of (1.22) actually vanishes because ${F^{\mu \nu }}$
is antisymmetric in its indices, whereas the second factor turns out
to be symmetric; see (1.29).