Surface Integrals

Consider an open surface S bounded by the closed curve C, as shown in Figure 2.9. At point P on the surface, let dS be an elemental area of the surface and n be a unit vector normal to the surface. The orientation of n is in the direction according to the right-hand rule for movement along C. (Curl the fingers of your right hand in the direction of movement around C; your thumb will then point in the general direction of n.) Define a vector elemental area as dS = ndS. In terms of dS, the surface integral over the surface S can be defined in three ways:

її p dS = surface integral of a scalar p over the 5 open surface S (the result is a vector)

If A • dS = surface integral of a vector A over the 5 open surface S (the result is a scalar)

N A x dS = surface integral of a vector A over the 5 open surface S (the result is a vector)

If the surface S is closed (e. g., the surface of a sphere or a cube), n points out of the surface, away from the enclosed volume, as shown in Figure 2.10. The surface integrals over the closed surface are