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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.
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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$:

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




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1999-07-01