  # 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

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

Step 2

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

Step 3

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### Integration 