2009-12-11: Theorems of Vector Calculus
Stewart section 17.8--17.9
- Key Point We now have a collection of theorems about vector integrals:
- The Fundamental Theorem of Calculus for line integrals,
- Green's Theorem,
- Stokes' Theorem, and
- the Divergence Theorem.
- It is most often that we use these to replace the integral on the left (a line or surface integral) with the expression on the right (a value, or a surface or volume integral) (though all, especially Stokes' theorem, may be used in either direction).
- Game: Let E be the
cylindrical volume with radius 3 and height 2, centered on the
z-axis and sitting on the xy-plane. Let C be
the circle around the base of the volume, and S be the
surface contained by C, and let F = <xy, y cos(z), x sin(z)>
- Find the flux over the surface of E.
- Find the line integral of F . dr on C using Stokes' theorem.
- How easy would it be to work out the line integral in (2) directly?
- Game: Let S_{1} be the sides of the cylinder, and S_{2} the top.
- Find the flux across S.
- Use (1) and the previous game to find the flux through S_{1}+S_{2}
- Find the flux through S
- Key Point
- Note that the flux integral of F is different from the flux integral of curlF.
- We can break up surfaces (and curves) to be able to use a theorem and an easy integral to find flux through (or the line integral on) part of the surface (or curve).
ma215-080-f09 lecture outline 2009-12-11
Created: Fri Dec 11 11:40:21 EST 2009
Comments to
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©2009 Gavin LaRose