166-169 All k × e https://wwwwrbavug points scakET8 169.504.XP Use the Divergenc

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166-169 All k × e https://wwwwrbavug points scakET8 169.504.XP Use the Divergence Theorem to calculate the surface integral Fs; that is, calculate the flux of F across s F(x, y, z) = x-z?j + 4xy'zk, s is the surface of the solid bounded by the cylinder 1 and the planes zx+ 10 and z . Need Help?RedtTalk to Tuter My Notes Ask Your 7.01 points 1 Previous Answers scakET8 16.7023 MI Evaluate the surface integral Fs for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F(x, y, z) = xy ? + yzj+zxk s is the part of the paraboloid z-2-xy that lies above the square 0 s xs 1, 0sy s 1, and has upward orientation Titer Need Help? Wareh Mater el O Type here to search

Explanation / Answer

Solution: 7

Write the surface z(x,y) as a vector:

S = xi + yj + z(x,y)k

dS = ?S/?x x ?S/?y dA = <1, 0, ?z/?x> x <0, 1, ?z/?y> dA

dS = <-?z/?x, -?z/?y, 1> dA (This is what dS always is when surface S = z(x,y) )

=> Let z = 2 - x2 - y2;

=> ?z/?x = -2x; and ?z/?y = -2y

dS = <-?z/?x, -?z/?y, 1> dA = <2x, 2y, 1> dx dy

??F•dS = ??<xy, yz, zx> • <2x, 2y, 1> dx dy, 0 ? x ? 1, 0 ? y ? 1
          = ??2x2y + 2y2z + xz dx dy, 0 ? x ? 1, 0 ? y ? 1
          = ??2x2y + 2y2[2 - x2 - y2] + x[2 - x2 - y2] dx dy, 0 ? x ? 1, 0 ? y ? 1
          = ??2x2y + 4y2 - 2x2y2 - 2y4 + 2x - x3 - xy2 dx dy, 0 ? x ? 1, 0 ? y ? 1
          = ? [2x3y/3 + 4xy2 - 2x3y2/3 - 2xy4 + x2 - x4/4 - x2y2/2]01 dy , 0 ? y ? 1
          = ? 2y/3 + 4y2 - 2y2/3 - 2y4 +1 - 1/4 - y2/2 dy , 0 ? y ? 1
          = ? 2y/3 + 17y2/6 - 2y4 +3/4 dy , 0 ? y ? 1
          = [y2/3 + 17y3/18 - 2y5/5 +3y/4]01
          = 1/3 + 17/18 - 2/5 +3/4
          = 293/180

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