Evaluate the surface integral: double integral F x dS 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)i + (yz)j + (zx)k, S is the part of the paraboloid z = 4 - x^2 - y^2 that lies above the square 0<=x<=1, 0<=y<=1, and has upward orientation.

Question
Asked Nov 10, 2019

Evaluate the surface integral: double integral F x dS 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)i + (yz)j + (zx)k, S is the part of the paraboloid z = 4 - x^2 - y^2 that lies above the square 0<=x<=1, 0<=y<=1, and has upward orientation. 

check_circleExpert Solution
Step 1

Given the vector field and paraboloid

vector field F - (xy)i +(yz)J+(zx)k
paraboloid z 4-x2-y
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vector field F - (xy)i +(yz)J+(zx)k paraboloid z 4-x2-y

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Step 2

Let g (x, y, z) be the point on surface S then gradient of g (x, y, z) is normal to S

g(x, y, 2) z-(4-x-y2
g(x, y, z) xy
+z-4
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g(x, y, 2) z-(4-x-y2 g(x, y, z) xy +z-4

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Step 3

Calculate the gradient of the...

og
Vg =
Vg 2xi +2yk
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og Vg = Vg 2xi +2yk

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Math

Calculus

Integration