may i know what mistake have i made?
given that the vector field F(x,y,z) = < -y,x,x-2y>. use stokes theorem to compute the upward flux of curl F across z=1-x^2 -y^2 with z>0.
what i did:
curl F =<-2,-1,2>
using stoke theorem, integral F. dr = double integral curl F dS = triple integral grad (Curl F) dxdydz =0 as grad (curl F)=0
may i know why this doesnt work?
using stoke theorem, integral F. dr = double integral curl F dS
this integral is along the closed curve dr, dS boundary is the closed curve dr.
dS may be arbitrary. (there is no volume)
double integral curl F dS = triple integral grad (Curl F) dxdydz =0
this integral is along volume dxdydz and surfase dS surounds this all volume.
does that mean that i cant convert it to triple integral because there is no volume?
sorry im still abit lost about why i cant convert it from double to triple integral. could you explain it to me again?
then when would i know when i can convert from single to double integrals and then from double to triple and when i should not convert ?
thanks so much for your help
You can't convert it to a triple integral because Stoke's theorem says nothing about triple integrals! Stoke's theorem says that a certain integral over a surface is equal to another integral around the boundary of that surface.