Let 
be an open, two-sided surface bounded by a closed non-intersecting curve 
(simple closed curve). Consider a directed line normal to as positive if it is on one side of , and negative if it is on the other side of . The choice of which side is positive is arbitrary but should be decided upon in advance. Call the direction or sense of positive if an observer, walking on the boundary of with his head pointing in the direction of the positive normal, has the surface on his left. Then if 
are single-valued, continuous, and have continuous first partial derivatives in a region of space including , we have
![\displaystyle \int_C[A_1dx + A_2dy + A_3dz] =\\\vspace{0.2in} \underset{S}{\iint}\left[ \left( \frac{\partial A_3}{\partial y} -\frac{\partial A_2}{\partial z} \right)\cos\alpha + \left( \frac{\partial A_1}{\partial z} -\frac{\partial A_3}{\partial x} \right)\cos\beta + \left( \frac{\partial A_2}{\partial x} -\frac{\partial A_1}{\partial y} \right)\cos\gamma \right]dS \cdots(38)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_C%5BA_1dx+%2B+A_2dy+%2B+A_3dz%5D+%3D%5C%5C%5Cvspace%7B0.2in%7D+%5Cunderset%7BS%7D%7B%5Ciint%7D%5Cleft%5B+%5Cleft%28+%5Cfrac%7B%5Cpartial+A_3%7D%7B%5Cpartial+y%7D+-%5Cfrac%7B%5Cpartial+A_2%7D%7B%5Cpartial+z%7D+%5Cright%29%5Ccos%5Calpha+%2B+%5Cleft%28+%5Cfrac%7B%5Cpartial+A_1%7D%7B%5Cpartial+z%7D+-%5Cfrac%7B%5Cpartial+A_3%7D%7B%5Cpartial+x%7D+%5Cright%29%5Ccos%5Cbeta+%2B+%5Cleft%28+%5Cfrac%7B%5Cpartial+A_2%7D%7B%5Cpartial+x%7D+-%5Cfrac%7B%5Cpartial+A_1%7D%7B%5Cpartial+y%7D+%5Cright%29%5Ccos%5Cgamma+%5Cright%5DdS+%5Ccdots%2838%29&bg=T&fg=000000&s=0 "\displaystyle \int_C[A_1dx + A_2dy + A_3dz] =\\\vspace{0.2in} \underset{S}{\iint}\left[ \left( \frac{\partial A_3}{\partial y} -\frac{\partial A_2}{\partial z} \right)\cos\alpha + \left( \frac{\partial A_1}{\partial z} -\frac{\partial A_3}{\partial x} \right)\cos\beta + \left( \frac{\partial A_2}{\partial x} -\frac{\partial A_1}{\partial y} \right)\cos\gamma \right]dS \cdots(38)")
In vector form with 
and 
, this is simply expressed as
In words this theorem, called Stoke’s theorem, states that the line integral of the tangential component of a vector 
taken around a simple closed curve is equal to the surface integral of the normal component of the curl of 
taken over any surface having as a boundary. Note that if, as a special case 
in (39), we obtain the result (28).
