Fi‎nd the flu‎x a‎cross the po‎sitively or‎iented su‎rface

Question

Fi‎nd the flu‎x of ^->F (x, y, z) =〈4x, 3z + x², y²/2〉

a‎cross the po‎sitively or‎iented su‎rface S giv‎en by

^->R(u, v) = 〈2u, 4v, −u²〉, 1 ≤ u² + v² ≤ 4,

Answer 1

The flux of F across the S is

∫∫_S F • dS = ∫∫_D F(R(u,v)) • R_u × R_v du dv

where D is the annulus in the (u,v)-plane determined by inequalities 1 ≤ u² + v² ≤ 4. Compute

R_u × R_v = <2, 0, -2u> × <0, 4, 0> = <8u, 0, 8>

F(R(u,v)) = F(2u, 4v, -u²) = <8u, u², 8v²>

Then

F(R(u,v)) • R_u × R_v = <8u, u², 8v²> • <8u, 0, 8>

...............................= (8u)(8u) + u²(0) + (8v²)(8) = 64u² + 64v²

................................= 64(u² + v²)

and hence ∫∫_D F(R(u,v)) • R_u × R_v du dv = 64 ∫∫_D (u² + v²) du dv. Using polar coordinates we find

64 ∫∫_D (u² + v²) du dv = 64 ∫₁² ∫₀^2π r² r dθ dr

...................................= 64 ∫₁² 2π r³ dr = 32π ∫₁² 4r³ dr

...................................= 32π (r⁴)|₁²

...................................= 32π (2⁴ - 1⁴)

...................................= 480π

Thus, the flux across S is 480π.