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) = xz i + x j + y k s is the hemisphere x2 + y2 + z2 = 81, y ≥ 0, oriented in the direction of the positive y-axis

Because I've gone ahead with trying to parameterize directly and learned the hard way that the resulting integral is large and annoying to work with, I'll propose a less direct approach.

Rather than compute the surface integral over straight away, let's close off the hemisphere with the disk of radius 9 centered at the origin and coincident with the plane . Then by the divergence theorem, since the region is closed, we have

where is the interior of . has divergence

so the flux over the closed region is

The total flux over the closed surface is equal to the flux over its component surfaces, so we have

Parameterize by

with and . Take the normal vector to to be

Then the flux of across is

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step-by-step explanation: