I am having trouble with this problem. If anyone could help that would be great. Let M be the capped cylindrical surface which is the union of two surfaces, a cylinder given by x^2+y^2=16, 0≤z≤1, and a hemispherical cap defined by x^2+y^2+(z−1)^2=16, z≥1. For the vector field F=(zx+z^2y+4y, z^3yx+3x, z^4x^2), compute ∬M(∇×F)⋅dS in any way you like.

Ok... I hope this is correct

Step-by-step explanation:

Let M be the capped cylindrical surface which is the union of two surfaces, a cylinder given by x^(2)+y^(2)=16

Center:  ( 0 , 0 )

Vertices:  ( 4 , 0 ) , ( − 4 , 0 )

Foci:  ( 4 √ 2 , 0 ) , ( − 4 √ 2 , 0 )

Eccentricity:  √ 2

Focal Parameter:  2 √ 2

Asymptotes:  y = x ,  y = − x

Then 0≤z≤1, and a hemispherical cap defined by x^2+y^2+(z−1)^2=16, z≥1.

Simplified

0 ≤ z ≤ 1 , x ^2 + y ^2 + z ^2 − 2 ^z + 1 = 16 , z ≥ 1

For the vector field F=(zx+z^2y+4y, z^3yx+3x, z^4x^2), compute ∬M(∇×F)⋅dS in any way you like.

Vector:

csc ( x )  ,  x = π

cot ( 3 x )  ,  x = 2 π 3

cos ( x 2 )  ,  x = 2 π

Since  

( z x + z ^2 y + 4 y , z ^3 y x + 3 x , z ^4 x ^2 )  is constant with respect to  F , the derivative of  ( z x + z ^2 y + 4 y , z ^3 y x + 3 x , z ^4 x 2 )  with respect to  F  is  0 .

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