haha
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haha asked in 科學數學 · 1 decade ago

Stokes's theorem F=(x-z)i+(y-

use Stokes's theorem to evalute ∫c F‧Tds

where C is the boundary of the part of the plane x+4y+z=12

lying in the first octant, and F=(x-z)i+(y-x)j+(z-y)k

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  • 1 decade ago
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    ∫c F‧Tds = ∫∫(▽xF)‧Ndσ (over region R)

    i   j   k

    ∂/∂x ∂/∂y ∂/∂z

    x-z  y-x  z-y

    this is a 3x3 matrix denoted it by M

    take det(M)

    then ▽xF = curl(F) = det(M) = (-1, -1, -1) is a vector

    ∫∫(▽xF)‧Ndσ= ∫∫(-1, -1, -1)‧(1, 4, 1)*[1/(1, 4, 1)‧(0, 0, 1)]*dxdy

    for N = (1, 4, 1) an upward-pointing normal vector

    =∫∫(-1 - 4 -1)*1/1*dxdy over region R

    the surface is the portion of the graph of z = 12-x-4y corresponding to (x, y) in the region R in the xy-plane bounded by the coordinate axes and the line x + 4y = 12

    = ∫∫(-6)dxdy where the area of the region R = 1/2*12*3 = 18

    = (-6)*18 = -108........Ans

    Source(s): me
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