Editing Covariant derivative (section)

==Informal definition using an embedding into Euclidean space==
Suppose an open subset {{mvar|U}} of a {{mvar|d}}-dimensional [[Riemannian manifold]] {{mvar|M}} is embedded into Euclidean space <math>(\R^n, \langle\cdot, \cdot\rangle)</math> via a [[Smoothness#Differentiability classes|twice continuously-differentiable]] (C{{sup|2}}) mapping <math>\vec\Psi : \R^d \supset U \to \R^n</math> such that the tangent space at <math>\vec\Psi(p)</math> is spanned by the vectors
<math display="block">\left\{ \left. \frac{\partial\vec\Psi}{\partial x^i} \right|_p : i \in \{ 1, \dots, d\}\right\}</math>
and the scalar product <math>\left \langle \cdot, \cdot \right \rangle </math> on <math>\R^n</math> is compatible with the metric on {{mvar|M}}:
<math display="block">g_{ij} = \left\langle \frac{\partial\vec\Psi}{\partial x^i}, \frac{\partial\vec\Psi}{\partial x^j} \right\rangle.</math>

(Since the manifold metric is always assumed to be regular,{{Clarify|date=August 2024|reason=regular means what in this context?}} the compatibility condition implies linear independence of the partial derivative tangent vectors.)

For a tangent vector field, {{nowrap|<math>\vec V = v^j \frac{\partial \vec\Psi}{\partial x^j}</math>,}} one has
<math display="block">\frac{\partial\vec V}{\partial x^i} = \frac{\partial}{\partial x^i} \left( v^j \frac{\partial \vec\Psi}{\partial x^j} \right)= \frac{\partial v^j}{\partial x^i} \frac{\partial\vec \Psi}{\partial x^j} + v^j \frac{\partial^2 \vec\Psi}{\partial x^i \, \partial x^j} .</math>

The last term is not tangential to {{mvar|M}}, but can be expressed as a linear combination of the tangent space base vectors using the [[Christoffel symbols]] as linear factors plus a vector orthogonal to the tangent space:
<math display="block"> v^j \frac{\partial^2 \vec\Psi}{\partial x^i \, \partial x^j} = v^j {\Gamma^k}_{ij} \frac{\partial\vec\Psi}{\partial x^k} + \vec n . </math>

In the case of the [[Levi-Civita connection]], the covariant derivative <math>\nabla_{\mathbf{e}_i} \vec V</math>, also written {{nowrap|<math>\nabla_i \vec V</math>,}} is defined as the orthogonal projection of the usual derivative onto tangent space:
<math display="block">
\nabla_{\mathbf{e}_i} \vec V := \frac{\partial\vec V}{\partial x^i} - \vec n = \left( \frac{\partial v^k}{\partial x^i} + v^j {\Gamma^k}_{ij} \right) \frac{\partial\vec\Psi}{\partial x^k}.
</math>

From here it may be computationally convenient to obtain a relation between the Christoffel symbols for the Levi-Civita connection and the metric. To do this we first note that, since the vector <math>\vec n</math> in the previous equation is orthogonal to the tangent space,
<math display="block">
\left\langle \frac{\partial^2 \vec\Psi}{\partial x^i \, \partial x^j}, \frac{\partial\vec \Psi}{\partial x^l} \right\rangle
= \left\langle {\Gamma^k}_{ij} \frac{\partial\vec\Psi}{\partial x^k} + \vec n, \frac{\partial\vec \Psi}{\partial x^l} \right\rangle
= \left\langle \frac{\partial\vec\Psi}{\partial x^k}, \frac{\partial\vec\Psi}{\partial x^l} \right\rangle {\Gamma^k}_{ij}
= g_{kl} \, {\Gamma^k}_{ij} .
</math>

Then, since the partial derivative of a component <math>g_{ab}</math> of the metric with respect to a coordinate <math>x^c</math> is
<math display="block">
\frac{\partial g_{ab}}{\partial x^c} = \frac{\partial}{ \partial x^c} \left\langle \frac{\partial \vec\Psi}{ \partial x^a}, \frac{\partial \vec\Psi}{\partial x^b} \right\rangle = \left\langle \frac{\partial^2 \vec\Psi}{ \partial x^c \, \partial x^a}, \frac{\partial \vec\Psi}{\partial x^b} \right\rangle + \left\langle \frac{\partial \vec\Psi}{\partial x^a}, \frac{\partial^2 \vec\Psi}{ \partial x^c \, \partial x^b} \right\rangle,
</math>

any triplet {{nowrap|<math>i, j, k</math>}} of indices yields a system of equations
<math display="block">
\left\{
\begin{alignedat}{2}
\frac{\partial g_{jk}}{\partial x^i} = &    & \left\langle \frac{\partial \vec\Psi}{\partial x^j}, \frac{\partial^2 \vec\Psi}{\partial x^k \partial x^i} \right\rangle & + \left\langle \frac{\partial \vec\Psi}{\partial x^k}, \frac{\partial^2 \vec\Psi}{\partial x^i \partial x^j} \right\rangle \\
\frac{\partial g_{ki}}{\partial x^j} = & \left\langle \frac{\partial \vec\Psi}{\partial x^i}, \frac{\partial^2 \vec\Psi}{\partial x^j \partial x^k} \right\rangle &    & + \left\langle \frac{\partial \vec\Psi}{\partial x^k}, \frac{\partial^2 \vec\Psi}{\partial x^i \partial x^j} \right\rangle \\
\frac{\partial g_{ij}}{\partial x^k} = & \left\langle \frac{\partial \vec\Psi}{\partial x^i}, \frac{\partial^2 \vec\Psi}{\partial x^j \partial x^k} \right\rangle & + \left\langle \frac{\partial \vec\Psi}{\partial x^j}, \frac{\partial^2 \vec\Psi}{\partial x^k \partial x^i} \right\rangle &    & .
\end{alignedat}
\right.
</math>
(Here the symmetry of the scalar product has been used and the order of partial differentiations have been swapped.)

Adding the first two equations and subtracting the third, we obtain
<math display="block">
\frac{\partial g_{jk}}{\partial x^i} + \frac{\partial g_{ki}}{\partial x^j} - \frac{\partial g_{ij}}{\partial x^k} =
2\left\langle \frac{\partial\vec \Psi}{\partial x^k}, \frac{\partial^2 \vec\Psi}{\partial x^i \, \partial x^j} \right\rangle.
</math>

Thus the Christoffel symbols for the Levi-Civita connection are related to the metric by
<math display="block">
g_{kl} {\Gamma^k}_{ij} = \frac{1}{2} \left( \frac{\partial g_{jl}}{\partial x^i} + \frac{\partial g_{li}}{\partial x^j}- \frac{\partial g_{ij}}{\partial x^l}\right).
</math>

If {{mvar|g}} is nondegenerate then <math> {\Gamma^k}_{ij} </math> can be solved for directly as

<math display="block">
{\Gamma^k}_{ij} = \frac{1}{2} g^{kl} \left( \frac{\partial g_{jl}}{\partial x^i} + \frac{\partial g_{li}}{\partial x^j}- \frac{\partial g_{ij}}{\partial x^l}\right).
</math>

For a very simple example that captures the essence of the description above, draw a circle on a flat sheet of paper. Travel around the circle at a constant speed. The derivative of your velocity, your acceleration vector, always points radially inward. Roll this sheet of paper into a cylinder. Now the (Euclidean) derivative of your velocity has a component that sometimes points inward toward the axis of the cylinder depending on whether you're near a solstice or an equinox. (At the point of the circle when you are moving parallel to the axis, there is no inward acceleration. Conversely, at a point (1/4 of a circle later) when the velocity is along the cylinder's bend, the inward acceleration is maximum.) This is the (Euclidean) normal component. The covariant derivative component is the component parallel to the cylinder's surface, and is the same as that before you rolled the sheet into a cylinder.