Stewart section 17.7

- Last time we looked at the
**area element***dS*for a**parametric surface***S*. -
**Key Point**Note what this says:**r**(*u*,*v*) maps a (*u*,*v*) pair to a point (*x*,*y*,*z*) in space. -
**Key Point**Thus int_{S}*dS*= surface area (an integral on*S*) = int_{D}|**r**_{u}x**r**_{v}|*du**dv*(an integral in the*u*-*v*plane). - Note also that
**r**_{u}and**r**_{v}are tangent to*S*, so their cross-product is perpendicular to it. -
**Game:**- Consider the surface
**r**= <*u*sin(*v*),*u*cos(2*v*), 2*v*>, 0<=*u*<=3, 0<=*v*<=pi. Set up an integral for its area. - If its density is
*d*=*z*, set up an integral for its mass.

- Consider the surface
- If
*f*(*x*,*y*,*z*) is a quantity associated with surface area (e.g., density), then the integral of*f*(**r**)*dS*adds this up over the surface. - Next: Recall that the integral of
**F**.*d***r**says "add up the piece of**F**that points along*C*." - For surfaces, a similar quentity is "the piece of
**F**that passes through the surface." This is called the**Flux**. -
**Game:**For*S*given by**r**= <*u*cos(*v*),*u*sin(*v*),*u*> (0 <=*u*<= 3, 0 <=*v*<= pi) and**F**= <-*y*,*x*,*z*>:- Find a vector perpendicular to
*S*. - Find a unit vector normal to
*S*. - Find the component of
**F**through*S*.

- Find a vector perpendicular to

Created: Mon Nov 30 13:35:44 EST 2009

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©2009 Gavin LaRose