Surface Integrals    (click here for discussion of surface integrals with oriented surfaces)

 

 

 

 

 

 

1.

 

In a Nut Shell:  Two questions are relevant.  What is a surface integral and how do

you evaluate a surface integral?

 

Recall the line integral, I, provides the value of a function, f(x,y,z), evaluated along

a curve, C, in space.  Here the line integral is       I  =  ∫ f(x, y, z) ds 

                                                                                      C

where   ds  is the arc length along the curve, C.   The surface integral, Is, is analogous

to the line integral in that it provides the value of a function, f(x,y,z), evaluated over a

“smooth” surface, S, in space.  Here the surface integral is  Is   =     ∫  ∫ f(x, y, z) dS 

                                                                                                             S

where   dS  is the element of surface area on the spatial surface.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2.

 

 

 

How do you evaluate a surface integral?

 

Usually the surface, S, in space is somewhat complicated.  So it is usually convenient

to transform the element of surface, dS, from the x-y-z space into a parallelogram

dA = du dv  in the u-v plane.           

where  r(u,v) is the parametric representation of the surface, S.

 

r(u,v)  =  <  x(u,v), y(u,v), z(u,v) >   is the position vector to point on surface, S

 

Click here to continue with discussion of surface integrals.

 


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