Let M be the capped cylindrical surface which is the union of two surfaces, a cylinder given by x2+y2=64, 0≤z≤1, and a hemispherical cap defined by x2+y2+(z−1)2=64, z≥1. For the vector field F=(zx+z2y+9y, z3yx+8x, z4x2), compute ∬M(∇×F)⋅dS in any way you like.

Stokes' theorem is an importent for determining the surface integral . In this theorem the flux of curl of the vector field can be calculated using the line integral of the vector field on the contour curve. Stokes' Theorem states:

∬ (∇ × F)ds = ∫ F.dr

S C

Note that given surface area is bounded by cylinder, x²+y² = 64 in xy plane.

We parameterise the surface S , r(t) = (x(t),y(t),z(t))

Let, x(t) = r cost , y(t) = r sint , z(t) = 0

=> x(t) = 8 cost , y(t) = 8 Sint , z(t) = 0

Therefore, r(t) = ( 8 cost , 8 sint , 0 )

Differentiating this , we have

r'(t) = ( -8 sint , 8 cost , 0)

F(r(t)) = ( 0× 8 cost + (0)( 8 sint ) + 9(8 sint), 0³( 8 sint )(8 cost) + 8(8 cost ) , 0⁴(8cost)²)

= ( 72 sint , 64 cost , 0)

Therefore, ∬ (∇ × F)ds = ∫ F.dr

S C

= ∫ (F (r(t)).r'(t) dt , 0≤ t ≤2π

= ∫(72 sint ,64 cost ,0).(-8 sint ,8 cost ,0)dt ,0≤t≤ 2π

= ∫(-576 sin²t , 512cos²t )dt , 0≤t≤2π

= ∫(-576( (1 - cos2t )/2) + 512( (1 + cos2t)/2))dt,

0≤t≤2π

= ∫(-288 (1 -cos2t) + 256(1 + cos2t)) dt, 0≤t≤2π

= ∫(-288 + 288cos2t + 256 + 256cos2t)dt, 0≤t≤2π

= ∫(-32 + 544cos2t ) dt , 0≤t≤2π

= [ -32 t + 544sin2t/2 ], 0<t<2π

= [ -32(2π) + 272 sin2(2π) ] ( sin(2π) = 0 )

= -64π

So, required value is -64π.

To learn more about Stock's theorem, refer:

https://brainly.com/question/14094270

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