Stokes' theorem

Consider a closed contour $l$ spanned by an open regular surface $S$, as shown in Fig. A.3. The positive direction along the path $l$ is linked to the positive direction for the unit normal $\hat{n}$ at any point on $S$by the so-called corkscrew rule: if the handle of a right-handed corkscrew were to rotate in the positive direction along $l$, then the screw would enter the cork in the positive $\hat{n}$ direction. Another way of linking the positive unit vector $\hat{l}$ along the path $l$ to the positive $\hat{n}$ on $S$ is by the so-called right hand rule: hold the thumb, index and middle fingers of your right hand at right angles to one another, as shown in Fig. ; if the index finger rotates to move parallel to the middle finger, and if in so doing its tip follows the contour $l$ in the positive direction, then the thumb is pointed in the $\hat{n}$ direction.

Figure A.3: Geometry for Stokes' theorem.

Stokes' theorem equates the flux of the curl of a vector field $\underline{f}$ across the open surface $S$ to the integral of $\underline{F} \cdot \hat{l}$ along the closed path $l$ which bounds $S$:

$$\int_S \nabla \times \underline{F} \cdot \hat{n} \, dS=\oint_l \underline{F} \cdot \hat{l} dl \space .$$ (2.16)