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Surface Integrals with “oriented surfaces”

In a Nut Shell: Surface integrals also appear in vector form

∫ ∫ F · dS = ∫ ∫ F · n dS Note: dS = n dS

S S

They involve the “dot product” of a vector function, F = F(x,y, z) with n the

unit normal to the surface, S. In such cases the unit normal to the surface may point

out (say the top of the surface) or may point in (say the bottom of the surface).

Such a surface, S, is said to be “orientable”. The direction of the unit vector, n,

establishes the orientation of the surface.

This situation is especially true for vector fields distributed over surfaces such as

flux passing through a surface, S, bounded by a spatial curve, C.

Definitions:

F = vector field (one example is fluid velocity giving rise to flux across a surface)

dS element of oriented surface S where dS = n dS

n = unit vector normal to oriented surface

as before the unit vector comes from the cross product n = (r_u x r_v) / | r_u x r_v |

where r is the position vector to the surface, S, given by

r = < x, y, z > = < x(u,v), y(u,v), z(u,v) > = r(u,v)

∫ ∫ F · dS = ∫ ∫ F · n dS = ∫ ∫ F · (r_u x r_v) / | r_u x r_v | dS

S S S

and recall dS = | r_u x r_v |dA where | r_u x r_v | transforms the element of area, dS,

on the surface, S, to the element of area, dA, on the uv-surface in the domain, D.

So the surface integral becomes

∫ ∫ F · dS = ∫ ∫ F · (r_u x r_v) dA

S D

Note that the dot product, F · (r_u x r_v) , is a scalar function and therefore it is

a scalar field. So the same two approaches, the direct approach and the

transformation approach, still apply to evaluate the scalar form of surface

integrals starting with this vector form of surface integral.

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