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

center= (-2,0)

radius= sqrt2/2

step-by-step explanation:

median is the value separating the higher half of a data sample, a population or probablity distribution from the lower half.

step-by-step explanation:

say you have 3,8,7,3,1,9,8 order them from lowest to highest

if orderd it would be 1,3,3,7,8,8,9 now you take away from the lowest and the highest until you end up in the middle.

so you would remove the 1 and 9 leaving you with 5 digits remaining

when you've reached the middle that would be your answer so in this case the answer is 7

if there were two numbers left say 7 and 5 you'd and them together so 12 then divide by 2 witch would equal 6 so the answer would be 6

good luck on that project hoped i you at all