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 - x^{2} - z^{2}, y>=0, and the vector field F = <-y, 1, x>.
- Game:
- Parameterize this surface as r(u,v) (use u=x, v=z).
- What is the (u,v) domain D?
- Find dS.
- 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 (m^{3}/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.
- Parameterize S as r(x,y).
- What is the (x,y) domain D?
- Find dS.
- Find F.dS on S.
- Find the flux through S.
ma215-080-f09 lecture outline 2009-12-02
Created: Wed Dec 2 11:52:52 EST 2009
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©2009 Gavin LaRose