I) We start with a non-Abelian Wilson line $^1$  

$$ \Psi(C)~:=~ P e^{\int_{C} \!A}  \tag{7.1} $$

over a parametrized (possibly open) curve $C$. Here $P$ denotes [path-ordering](http://en.wikipedia.org/wiki/Path-ordering). 

II) We now make an infinitesimal variation of the curve $C$ to a new curve $C^{\prime}$. We may define an infinitesimally thin 2-surface $\Sigma$ with oriented boundary $^2$

$$ \partial \Sigma~=~ C^{\prime}-C \tag{A}$$

given by the two curves $C$ and $C^{\prime}$.

III) One may show a non-Abelian version of [Stokes' circulation theorem](http://en.wikipedia.org/wiki/Stokes%27_theorem) $^3$

$$\begin{align}  \delta\!\int_{C}\! A ~:=~&\int_{C^{\prime}}\! A-\int_{C} \!A\cr ~=~& \oint_{\partial\Sigma} \! A~=~ \iint_{\Sigma}\! F . \end{align}\tag{B} $$

The sketched proof of the  non-Abelian eq. (B) is in two steps as its Abelian cousin:

1. Split the $2$-surface $\Sigma$ into infinitely many infinitesimal small polygons, where contributions from internal edges of the polygons cancel because of opposite orientations.

2. Notice that for a sufficiently small polygon the holonomy around the polygon is approximately equal to the non-Abelian field strength $F$ times the area.

IV) Now we are ready to evaluate the (passive) change in holonomy because of the change $C\to C^{\prime}$ in curves:

$$\begin{align} \delta\Psi(C)~:=~&\Psi(C^{\prime})-\Psi(C)\cr ~=~& P\left( e^{\int_{C} \!A} ~\delta\!\int_{C} \!A \right)\cr ~\stackrel{(B)}{=}~& P\left( e^{\int_{C} \!A}   ~\iint_{\Sigma}\! F\right).\end{align}\tag{7.25} $$

Because of the path-ordering $P$ in eq. (7.25), we are strictly speaking applying a "chopped up version" of eq. (B) consisting of infinitely many infinitesimally small polygons, which reflect the proof technique of eq. (B).

References:

1. A.M. Polyakov, _Gauge Fields and Strings,_ 1987.


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$^1$ A Wilson line $\Psi(C)$ is physics jargon for [holonomy](http://en.wikipedia.org/wiki/Holonomy). If the curve $C$ is closed, we speak of a Wilson loop rather than a Wilson line. 

$^2$ For an open curve $C$, the varied curve $C^{\prime}$ is assumed to have the same end points as $C$. We let both curves $C$ and $C^{\prime}$ be parametrized so that we have a common notion of path-ordering $P$.

$^3$ Let us mention for completeness that the exponentiated version 

$$ \tag{C} P\exp\oint_{\partial\Sigma} \! A~=~ P_2\exp\iint_{\Sigma}\! F $$

of [non-Abelian Stokes' Theorem](http://ncatlab.org/nlab/show/nonabelian+Stokes+theorem) depends on a choice of surface ordering $P_2$. Note that an infinitesimal thin 2-surface $\Sigma$ has a natural choice of surface ordering $P_2$.