W7V2 - Orientation and Flux PDF

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Orientation and Flux We say a surface S parameterized by ~r (u, v) has orientation ~n =

~ru × ~rv k~ru × ~rv k

Find the orientation of the following two parameterizations of the unit sphere: ~r (u, v) = [cos(u) sin(v), sin(u) sin(v), cos(v)], 0 ≤ u ≤ 2π, 0 ≤ v ≤ π

~p(u, v) = [cos(v) sin(u), sin(v) sin(u), cos(u)], 0 ≤ v ≤ 2π, 0 ≤ u ≤ π

We say that ~p has positive (outward) orientation and ~r has negative (inward) orientation.

This matters because besides integrating real valued functions, we can also integrate vector fields over surfaces. We define the flux of F~ over the surface S to be the surface integral of F~ · ~n. (where it is implied that S has outward orientation.)

So flux is equal to

ZZ

~= F~ · dS S

ZZ

~ · ~ndS = F S

ZZ

~ (r(u, v)) · (~ru × ~rv )dA F D...