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Stokes' Theorem

Stokes' theorem (which might well be called the curl theorem if we wanted to be more consistent in our terminology) is equally critical to our future work:

$\displaystyle \int_{S/C} (\curl \vV)\cdot \hn dA = \oint_C \vV \cdot d\vell

Again, read $ S/C$ as ``the open surface $ S$ bounded by the closed curve $ C$ , and note that there is an implicit direction in this equation. In particular, you must choose (from the two possible choices) the direction for $ \hn$ that corresponds to the right-handed direction around the loop $ C$ . In words, if you curl the fingers of your right hand around $ C$ in the direction in which you wish to do the integral, your thumb should point ``through'' the loop $ C$ in the direction you must select for the normal.

We can once again derive an additional form of the curl theorem/Stokes' theorem:

$\displaystyle \int_{S/C} (\hn \times \grad f )\cdot dA = \oint_C f d\vell

Note well that the $ \hn$ has been moved to the front!

Robert G. Brown 2017-07-11