- #1
gtfitzpatrick
- 379
- 0
Homework Statement
Use stokes theorem to elaluate to integral [itex] \int\int_{s} curlF.dS[/itex] where [itex] F(x,y,z)= x^2 z^2 i + y^2 z^2 j + xyz k [/itex] and s is the part of the paraboliod [itex]z=x^2+ y^2 [/itex] that lies inside the cylinder [itex] x^2 +y^2 =4 [/itex] and is orientated upwards
Homework Equations
The Attempt at a Solution
so i use Stokes theorem [itex] \int\int_{s} curlF.dS = \oint_{c} F.dv[/itex]
so i want to get parametric equations for C and so i get x=2cost y=2sint and z=4 i came up with these as the boundary curve c is a circle of raadius 2 and my z value of 4 because that is where the parabaloid and cylinder intersect...am i right in my thinking here?
so then dx= -2sintdt ; dy=2costdt and dz=0
so then i get [itex] \oint_{c} F.dv = \oint_{c} x^2 z^2 dx - y^2z^2 dy + xyzdz[/itex]
which becomes [itex]\int^{2\pi}_{0} ((4cos^2 t )(16)(-2sint) - (4sin^2t)(16)(ccost)) dt[/itex]
[itex]\int^{2\pi}_{0} ((-128cos^2 t )(sint) - (128sin^2 t)(cost)) dt[/itex]
am i working on the right lines here?
Last edited: