Evaluate the surface integral s f · 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 − x2 − y2 that lies above the square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, and has upward orientation

The value of the surface integral S F · dS for the given vector field F and the oriented surface S is 713/180 and this can be determined by doing integration.

Given :

First, S is parameterized in order to evaluate the surface integral.

 where and .

The expression for is given by:

Now, the surface integral is given by:

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Parameterize by

with and . Then

In the integral we get