Question

Use Stokes's theorem to calculate \(\iint_{S}(\nabla \times \boldsymbol{F}) \cdot d S\) where

$$ \boldsymbol{F}=(x-z) \boldsymbol{i}+\left(x^{3}+y z\right) \boldsymbol{j}-3 x y^{2} \boldsymbol{k} $$

and \(S\) is the surface of the cone \(z=2-\sqrt{x^{2}+y^{2}}\) above the \(x y\) plane.

Answer 1

Given z=2- $2²742 and xy-plane ul z=0 x² + y² = 4 (2-0) And F = (2-2) ? + (x² + y2) ŷ = 3x y z Ř The parameterization of this curve is, } (t) = 2 Cost êt2dint ŷ to R lie. g (t) = 2 cost êtasint Î ostega Now, E dr SS, (vxF), ds = S San F(74). 814) 4+ = Now, È (8 (A)) (200s-o) + (80994+ o) - (3)(2 cos +) (4.5in?+) where, d=alost, y=asint, 220 ² (12) = 2 lost î+ 8 Cos? t } - 24 Casa sin?t ê And 8! (t) =- asint i + 2 cost ŷ

Now, 27 27 Fot) . '(t) dd = [2lost i + 8 Cost -24 Cast Sinzt ř]. [-2sint + 2005 & f] dt 97 4 Cost Sint +1660544] (2x Cost sint dat +16 cha "Costadt - -4. 0 -45 [-4 cos**** 16 [( 12% = - 4 + 8 sinadt sin 4at = faltes 2005²t +62 +4 sin at the sinft = 12W*40+] - [8464040] [ 2+123] - [2] - 127