Verify the Divergence Theorem for F=(2xz,y,−z^2) and D is the wedge cut from the first octant by the plane z =y and the elliptical cylinder x^2+4y^2=16
∫∫F．n．dS=∫∫∫divF．dv
r(u,v)=(4cosu,2sinu,v) where u=[0,pi/2] and v=[0,1]
→ru×→rv=(2cosu,4sinu,0)
r→u×r→v=(2cosu,4sinu,0)
\(\displaystyle \int \int f(r(u,v))\cdot \overrightarrow{r}u \times \overrightarrow{r}v dudv \)
For the Div F ． dv
Div F=1
Using cylindrical coordinate
The limit is dz: 0<=z<=1 , dθ: 0<=θ<=pi/2
How can I find the dr limit for the integration?
since it is not a cylinder
thus,∫∫∫r dzdrdθ
Am I above step correct?
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[/FONT][FONT="]‖r→u×r→v‖=4cos2u+16sin2v[/FONT]