2009-12-07: More Stokes' Theorem
Stewart section 17.8
- Last time we introduced Stokes' theorem, and noted that it allows us to replace a surface integral with a line integral, or vice versa. Our goal is to evaluate whichever of these is the easiest.
- Note that we said that curl F is the tendency of the vector field F to rotate things. So Green's and Stokes' theorems both say that this tendency on a surface is captured by the behavior on the boundary!
- Example: Consider a surface S and F = <2x, x^{2}y, -xz>. Find the double integral on S of curl F dot dS.
- Game:
- The boundary of S consists of two parts. Find r(t) on each.
- Find the integral of F dot dr on the boundary.
- Note that the surface T [figure] has the same boundary as S.
- Game: Find the integral of curl F dot dS on T.
- Find dS (hint: note that dS = dx dz).
- Find curl F.
- Find the integral.
- Key Point
- dS = r_{u} x r_{v} du dv = N dS. Thus if N = i (or j or k), we can write down dS without parameterizing the surface!
- Stokes' theorem lets us evaluate either a surface integral or a line integral.
- It also lets us evaluate a surface integral on a different surface, if the second surface has the same boundary.
ma215-080-f09 lecture outline 2009-12-07
Created: Mon Dec 7 12:10:45 EST 2009
Comments to
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©2009 Gavin LaRose