Stokes theorem to calculate the circulation of field F?

Question

Stokes theorem to calculate the circulation of field F around the curve C in the indicated direction

F= 2yi + yj + zk; C the counterclockwise path boundary of the ellipse x^2/49 + y^2/16 = 1

Thank you!!!!

Answer 1

The boundary is missing a z-value (from the surface or something).

Assuming that the ellipse lies on the xy-plane, I'll assume that z = 0.



By Stokes' Theorem, ∫∫s curl F · dS = ∫c F · dr.



Parameterizing C via r(t) = <7 cos t, 4 sin t, 0> for t in [0, 2π],

∫c F · dr

= ∫(t = 0 to 2π) <2 * 4 sin t, 4 sin t, 0> · <-7 sin t, 4 cos t, 0> dt

= ∫(t = 0 to 2π) (-56 sin^2(t) + 16 sin t cos t) dt

= ∫(t = 0 to 2π) (-28 (1 - cos(2t)) + 16 sin t cos t) dt

= ∫(t = 0 to 2π) (-28 + 28 cos(2t) + 16 sin t cos t) dt

= (-28t + 14 sin(2t) + 8 sin^2(t)) {for t = 0 to 2π}

= -56π.



I hope this helps!

Answer 2

Stokes Theorem Circulation

Answer 3

You meant Gauss theorem .- on the grounds that F.ok =0 , the flux on the caps are =0 and Gauss will provide us the flux over the shell .- Flux= INT_S F.dS = INT_C Div F dV ( Gauss) Div F=0 , so flux=0