# The parallel transporter in terms of the connection

We can also consider the parallel transport of a vector $${w}$$ along an infinitesimal curve $${C}$$ with tangent $${v}$$. Referring to the previous figure, we see that to order $${\varepsilon}$$ the components $${w^{\mu}}$$ transform according to

$$\displaystyle \parallel^{\lambda}{}_{\mu}\left(C\right)w^{\mu}=w^{\lambda}-\varepsilon\Gamma^{\lambda}{}_{\mu}\left(v\right)w^{\mu},$$

where $${v}$$ is tangent to the curve $${C}$$, and these components are with respect to the frame at the new point after infinitesimal parallel transport. Using this relation, we can build up a frame-dependent expression for the parallel transporter for finite $${C}$$ by multiplying terms $${\left(1-\varepsilon\Gamma\left|_{p}\right.\right)}$$ where $${\Gamma\left|_{p}\right.}$$ is used to denote the matrix $${\Gamma^{\lambda}{}_{\mu}\left(v\left|_{p}\right.\right)}$$ evaluated on the tangent $${v\left|_{p}\right.}$$ at successive points $${p}$$ along $${C}$$. The limit of this process is the path-ordered exponential

\begin{aligned}\parallel^{\lambda}{}_{\mu}\left(C\right) & =\underset{\varepsilon\rightarrow0}{\textrm{lim}}\left(1-\varepsilon\Gamma\left|_{q-\varepsilon}\right.\right)\left(1-\varepsilon\Gamma\left|_{q-2\varepsilon}\right.\right)\dotsm\left(1-\varepsilon\Gamma\left|_{p+\varepsilon}\right.\right)\left(1-\varepsilon\Gamma\left|_{p}\right.\right)\\ & \equiv P\textrm{exp}\left(-\underset{C}{\int}\Gamma^{\lambda}{}_{\mu}\right),
\end{aligned}

whose definition is based on the expression for the exponential

$$\displaystyle e^{x}=\underset{n\rightarrow\infty}{\textrm{lim}}\left(1+\frac{x}{n}\right)^{n}=\underset{\varepsilon\rightarrow 0}{\textrm{lim}}\left(1+\varepsilon x\right)^{1/\varepsilon}.$$

Note that the above expression for $${\parallel^{\lambda}{}_{\mu}\left(C\right)}$$ exponentiates frame-dependent values in $${gl\left(n,\mathbb{R}\right)}$$ to yield a frame-dependent value in $${GL\left(n,\mathbb{R}\right)}$$.