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## 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

Question

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.

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Mathematics
6 months
2021-07-27T16:06:49+00:00
2021-07-27T16:06:49+00:00 1 Answers
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## Answers ( )

Answer: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

.