Calculus

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find the flux of F=<0,y,z> across the parabolic sheet given by x=2y^2+2z^2+3, with x <= 5

  • Calculus -

    Flux = I = ∫∫F.n dS
    for S over the surface, g(x,y,z), where
    g(x,y,z) is the part of
    x=2y²+2z²+3 where x≤5, which translates to
    g(x,y,z)=x-2y²-2z²-3
    and the region is the part of the paraboloid where y²+z²≤1 after substitution of x≤5.
    This region is circular with radius =1, which facilitates integration later.

    Now:
    F = <0,y,z>
    n=∇g/||g||
    so I can be evaluated:
    I = ∫∫F.n dS
    The surface integral can be evaluated by projecting it to the region R on the y-z plane by multiplying:
    dS over S = ||g|| dA over R

    So
    I = ∫∫F.n dS over S
    = ∫∫F.n ||g|| dA over R
    where R is the circular region r≤1 in polar coordinates.

    The way I have defined g(x,y,z) results in negative flux.

    Post if you could use more help.

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