Flux of a vector field.

31. FluxLet S ⊂ R3 be a smooth 2-manifold and let�g : U −→ S ∩ W, be a diffeomorphism. Definition 31.1. Let f : S ∩ W −→ R and F�: S ∩ W −→ R3 be two functions, the first a scalar function and the second a vector field. We define �� �� ∂�g ∂�gf dS = f(g(s,t))� ∂s × ∂t � ds dt �� S∩W �� S∩W �� F�dS�= F�(g(s,t)) ∂�g ∂�g ds dt.·· ∂s × ∂t S∩WS∩W The second integral is called the flux of F�across S in the direction of ∂�g ∂�g nˆ= ∂s × ∂t . �∂�g ∂�g ∂s × ∂t � Note that �� ��F�dS�=(F�ˆS.n)d�·· S∩WS∩W Note also that one can define the line integral of f and F�over the whole of S using partitions of unity. Example 31.2. Find the flux of the vector field given by F�(x,y,z)= yˆj + xk, ı + zˆˆthrough the triangle S with vertices P0 = (1, 2, −1) P1 = (2, 1, 1) and P2 = (3, −1, 2), in the direction of −−→−−→P0P1 × P0P2 nˆ= . �−−→P0P1 × −−→P0P2�First we parametrise S,�g : U −→ S ∩ W, where g(s,t)= −−→P0P1 + t−−→OP0 + s−−→P0P2 = (1+ s +2t, 2 − s − 3t, −1+2s +3t), and U = { (s,t) ∈ R2 0