# 2009-12-02: Surface Integrals: Flux Integrals

Stewart section 17.7

• Key Point In the same way that the line integral F dot dr adds up the magnitude of F in the direction of a curve C times arclength, the surface integral F dot dS adds up the magnitude of F in the direction perpendicular to the surface times the surface area.
• Consider the surface y = 9 - x2 - z2, y>=0, and the vector field F = <-y, 1, x>.
• Game:
1. Parameterize this surface as r(u,v) (use u=x, v=z).
2. What is the (u,v) domain D?
3. Find dS.
4. Find F.dS on the surface S.
• After finding F.dS, we can find the total flux through the surface.
• What is the meaning of Flux? If F is a velocity, then F.dS is a volumetric flow rate (m3/s): the flux is the rate the velocity field moves material through the surface.
• Game: Let F = <-z, 1, x> and S be the plane x+y+z=1, in the first octant.
1. Parameterize S as r(x,y).
2. What is the (x,y) domain D?
3. Find dS.
4. Find F.dS on S.
5. Find the flux through S.
ma215-080-f09 lecture outline 2009-12-02
Created: Wed Dec 2 11:52:52 EST 2009