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:

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

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

Note well that the has been moved to the front!

Robert G. Brown 2017-07-11