A.k.a. Gauss's Theorem. Let be a region in space with boundary . Then

(1) 
Let be a region in the plane with boundary .

(2) 
If the Vector Field satisfies certain constraints, simplified forms can be used. If
where is a constant vector , then

(3) 
But

(4) 
so

(5) 

(6) 
But
, and
must vary with so that
cannot
always equal zero. Therefore,

(7) 
If
, where is a constant vector , then

(8) 
See also Curl Theorem, Gradient, Green's Theorem
References
Arfken, G. ``Gauss's Theorem.'' §1.11 in
Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 5761, 1985.
© 19969 Eric W. Weisstein
19990524