Editing Covariant derivative (section)

==Coordinate description==
{{Hatnote|This section uses the [[Einstein summation convention]].}}
Given coordinate functions <math display="block">x^i,\ i=0,1,2,\dots ,</math> any [[tangent vector]] can be described by its components in the basis <math display="block">\mathbf{e}_i = \frac{\partial}{\partial x^i} .</math>

The covariant derivative of a basis vector along a basis vector is again a vector and so can be expressed as a linear combination <math>\Gamma^k \mathbf{e}_k</math>.
To specify the covariant derivative it is enough to specify the covariant derivative of each basis vector field <math>\mathbf{e}_i</math> along <math>\mathbf{e}_j</math>.
<math display="block"> \nabla_{\mathbf{e}_j} \mathbf{e}_i = {\Gamma^k}_{i j} \mathbf{e}_k,</math>

the coefficients <math>\Gamma^k_{i j}</math> are the components of the connection with respect to a system of local coordinates. In the theory of Riemannian and pseudo-Riemannian manifolds, the components of the Levi-Civita connection with respect to a system of local coordinates are called [[Christoffel symbols]].

Then using the rules in the definition, we find that for general vector fields <math>\mathbf{v} = v^j \mathbf{e}_j </math> and <math>\mathbf{u} = u^i \mathbf{e}_i</math> we get
<math display="block">\begin{align}
 \nabla_\mathbf{v} \mathbf{u}
&= \nabla_{v^j \mathbf{e}_j} u^i \mathbf{e}_i \\
&= v^j \nabla_{\mathbf{e}_j} u^i \mathbf{e}_i \\
&= v^j u^i \nabla_{\mathbf{e}_j} \mathbf{e}_i + v^j \mathbf{e}_i \nabla_{\mathbf{e}_j} u^i \\
&= v^j u^i {\Gamma^k}_{i j}\mathbf{e}_k + v^j{\partial u^i \over \partial x^j} \mathbf{e}_i
\end{align}</math>

so
<math display="block"> \nabla_\mathbf{v} \mathbf{u} = \left(v^j u^i {\Gamma^k}_{i j} + v^j {\partial u^k\over\partial x^j} \right)\mathbf{e}_k .</math>

The first term in this formula is responsible for "twisting" the coordinate system with respect to the covariant derivative and the second for changes of components of the vector field {{mvar|u}}. In particular
<math display="block">\nabla_{\mathbf{e}_j} \mathbf{u} = \nabla_j \mathbf{u} = \left( \frac{\partial u^i}{\partial x^j} + u^k {\Gamma^i}_{kj} \right) \mathbf{e}_i </math>

In words: the covariant derivative is the usual derivative along the coordinates with correction terms which tell how the coordinates change.

For covectors similarly we have
<math display="block">\nabla_{\mathbf{e}_j} {\mathbf \theta} = \left( \frac{\partial \theta_i}{\partial x^j} - \theta_k {\Gamma^k}_{ij} \right) {\mathbf e^*}^i </math>

where <math>{\mathbf e^*}^i (\mathbf{e}_j) = {\delta^i}_j</math>.

The covariant derivative of a type {{math|(''r'', ''s'')}} tensor field along <math>e_c</math> is given by the expression:

<math display="block">\begin{align}
{(\nabla_{e_c} T)^{a_1 \ldots a_r}}_{b_1 \ldots b_s} = {}
&\frac{\partial}{\partial x^c}{T^{a_1 \ldots a_r}}_{b_1 \ldots b_s} \\
&+ \,{\Gamma ^{a_1}}_{dc} {T^{d a_2 \ldots a_r}}_{b_1 \ldots b_s} + \cdots + {\Gamma^{a_r}}_{dc} {T^{a_1 \ldots a_{r-1}d}}_{b_1 \ldots b_s} \\
&-\,{\Gamma^d}_{b_1 c} {T^{a_1 \ldots a_r}}_{d b_2 \ldots b_s} - \cdots - {\Gamma^d}_{b_s c} {T^{a_1 \ldots a_r}}_{b_1 \ldots b_{s-1} d}.
\end{align}</math>

Or, in words: take the partial derivative of the tensor and add: <math>+{\Gamma^{a_i}}_{dc}</math> for every upper index <math>a_i</math>, and <math>-{\Gamma^d}_{b_ic}</math> for every lower index <math>b_i</math>.

If instead of a tensor, one is trying to differentiate a ''[[tensor density]]'' (of weight +1), then one also adds a term
<math display="block">-{\Gamma^d}_{d c} {T^{a_1 \ldots a_r}}_{b_1 \ldots b_s}.</math>
If it is a tensor density of weight {{mvar|W}}, then multiply that term by {{mvar|W}}.
For example, <math display="inline"> \sqrt{-g}</math> is a scalar density (of weight +1), so we get:
<math display="block">\left(\sqrt{-g}\right)_{;c} = \left(\sqrt{-g}\right)_{,c} - \sqrt{-g}\,{\Gamma^d}_{d c}</math>

where semicolon ";" indicates covariant differentiation and comma "," indicates partial differentiation. Incidentally, this particular expression is equal to zero, because the covariant derivative of a function solely of the metric is always zero.