University Mathematics Handbook (2015)
IX. Vector Analysis
Chapter 10. Surface Integral
10.1 Surfaces
a. is a smooth surface if vector function is of class region.
b. The unit normal to surface is
c. A smooth surface S is called oriented (two-sided) if there is a normal unit vector n at every point on S, not on the boundary curve of S, such that n is a continuous function of (x, y, z) on S. Later on, each surface must be oriented, have a finite area and a single smooth, closed boundary curve. The sphere, ellipsoid, paraboloid, and hyperboloid are two-sided surfaces.
A nonorientable surface is, for example, a Mobius strip. It is formed by half-twisting a strip of paper, attaching its ends together (see illustration). If we draw a unit normal N from one point of a Mobius strip, we eventually get a normal at the same point, at the opposite direction of the original normal.
d. A smooth surface is two-sided if the direction of normal can on it can be specified definitely. Later on, we shall only refer to two-sided surfaces.
e. The area of the surface is .
f. If surface is given in the form , then its area is .
g. The positive direction of surface is the direction of normal . If is the boundary of a closed body, then the positive direction on is outwards. If is an open surface, then the positive direction is towards the Z-axis.
10.2 Surface Integral
Given a vector field
a. is a surface integral when is the unit normal vector to surface .
b. Different ways of writing and calculating surface integral:
when are directional angles of .
c. Gauss Divergent Theorem
Let be a body the boundary of which is smooth and closed surface with an outer normal . If vector field is of class in , then holds
or .
The integral is called the flux of vector field through surface .
d. Stokes Theorem
If, in the neighborhood of two-sided surface , vector field is of class and is the boundary of surface in the positive direction, then
or
denote