A vector field in ${\mathbb{R}}^n$ can be seen as a differentiable ($C^{\infty}$ ) map $V\colon{\Bbb{R}}^n\to {\mathbb{R}}^n$ .

Or as a section ${\mathbb{R}}^n\stackrel{V}\to T({\mathbb{R}}^n)$ where $T{\mathbb{R}}^n\equiv{\mathbb{R}}^n\times{\mathbb{R}}^n$ is the ${\mathbb{R}}^n$ 's trivial tangent bundle obeying $p\mapsto (p,V(p)\in T_p({\mathbb{R}}^n))$ with $T_p({\mathbb{R}}^n)\equiv{\mathbb{R}}^n$ being the tangent space at $p$ .

Another viewpoint about tangent vectors is that they are also linear operators called derivations and they act over scalars $f\colon {\mathbb{R}}^n\to{\mathbb{R}}$ via $p\mapsto Vf|_p=V(p)\cdot\nabla f|_p$ .

Let $X$ be one of them and $dX|_p$ its Jacobian matrix evaluated at the point $p\in{\Bbb{R}}^n$ . Then, for any other vector field $Y\colon{\mathbb{R}}^n\to{\mathbb{R}}^n$ , $$dX|_p(Y(p))$$ measures how $X$ varies in the direction $Y$ at $p$ .

We have $dX|_p(Y(p))=(Y(p)\cdot\nabla X^1|_p,...,Y(p)\cdot\nabla X^n|_p)$ , where $X=\sum_sX^se_s$ in components. Also, it is obvious that $p\mapsto dX|_p(Y(p))$ defines a new vector field in ${\mathbb{R}}^n$ which is symbolized as $$D_YX$$ We can be consider it as a bilinear map $$D:T({\mathbb{R}}^n)\times T({\mathbb{R}}^n)\to T({\mathbb{R}}^n).$$ $$(X,Y)\mapsto D_XY$$ Further, it is easy to see that for any scalar $f\colon {\mathbb{R}}^n\to{\mathbb{R}}$

1. $D_{fY}X=fD_YX$
2. $D_Y(fX)=(Yf)X+ fD_YX$
3. $D_XY-D_YX=[X,Y]$
4. $X(Y\cdot Z)=D_XY\cdot Z+X\cdot D_XZ$

This $D$ is called the standard connection of ${\mathbb{R}}^n$ .

Now, let $M$ be a n-dimensional differentiable manifold and let $TM$ be its tangent bundle. The set of differentiable sections $\Gamma(M)=\{X\colon M\to TM\}$ is a differentiable Lie algebra which is endowed with a differentiable inner product $g\colon\Gamma(M)\times\Gamma(M)\to{\mathbb{R}}$ via $$g(X,Y)|_p=X(p)\cdot Y(p)$$ in each $T_p(M)\equiv{\mathbb{R}}^n$ .

It is possible construct a bilinear operator $\nabla$ $$\nabla\colon \Gamma(M)\times\Gamma(M)\to\Gamma(M)$$ compatible with $g$ and which satisfies the following properties

1. $\nabla_{fY}X=f\nabla_YX$
2. $\nabla_Y(fX)=(Yf)X+ f\nabla_YX$
3. $\nabla_XY-\nabla_YX=[X,Y]$
4. $Xg(Y,Z)=g(\nabla_XY,Z)+g(X,\nabla_XZ)$

The Fundamental Theorem of Riemannian Geometry establishes that this $\nabla$ exists and it is unique, and it is called the Levi-Civita connection for the metric $g$ on $M$ .

Now, if one uses a coordinated patch in $M$ one has a set of n-coordinated vector fields $\partial_1,..,\partial_n$ meaning $\partial_i={{\partial}\over{\partial u^i}}$ being $u^i$ the coordinate functions. These are also dubbed holonomic derivations.

So it makes sense to speak about the derivatives $\nabla_{\partial_i}\partial_j$ and since the $\partial_i$ are tangent which generate at a point $T_p(M)$ , then $\nabla_{\partial_i}\partial_j$ is also tangent, so there are $n\times n$ numbers (functions if one varies position) $\Gamma^s_{ij}$ which enters in the relation $$\nabla_{\partial_i}\partial_j=\sum_s\Gamma^s_{ij}\partial_s.$$ These coefficients $\Gamma^s_{ij}$ are called Christoffel symbols and an easy calculation shows that $$\Gamma^k_{ij}={1\over 2}\sum_sg^{ks}[g_{sj,i}+g_{is,j}-g_{ij,s}]$$ where $g_{ij}=g(\partial_i,\partial_j)$ , $g^{ij}$ are the entries of the matrix $[g_{ij}]^{-1}$ and $g_{ij,k}=\partial_k(g_{ij})$ .

Routinely one can check that under a change of coordinates $u^i\to w^j$ these functions transform as $$\bar{\Gamma}^i_{kl}= {{\partial w^i}\over{\partial u^m}}{{\partial u^n}\over{\partial w^k}}{{\partial u^p}\over{\partial w^l}} \Gamma^m_{np}+{{\partial}^2u^p\over{\partial w^k\partial w^l}}{{\partial w^i}\over{\partial u^p}} $$ here we have used Einstein's sum convention ($m,n,p$ -sums) and the term $${ {\partial}^2u^p\over{\partial w^k\partial w_l} } { {\partial w^i}\over{\partial u^p} }$$ shows that the $\Gamma^i_{kl}$ are not tensors.

For a proof please see the last part in: http://planetmath.org/?op=getobj&from=collab&id=64

Connection with base vectors.

Let us assume that coordinates $u^i$ are referred to a right-handed orthogonal Cartesian system with attached constant base vectors $\mathbf{e}_i\equiv\mathbf{e}^i$ and coordinates $w^j$ referred to a general curvilinear system attached to a local covariant base vectors $\mathbf{g}_j$ and local contravariant base vectors $\mathbf{g}^k$ , both systems embedded in the Euclidean space $\mathbb{R}^n$ . We shall also suppose diffeomorphic the transfomation $u^i\mapsto w^j$ . Then, by definition

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