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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Yến Oanh 6 months 2021-07-27T16:06:49+00:00 1 Answers 12 views 0

Answers ( )

    0
    2021-07-27T16:08:25+00:00

    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
    .

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