Let m be the closed surface that consists of the hemisphere
m1: x2 + y2 + z2 = 1, z = 0,
and its base
m2: x2 + y2 = 1, z = 0 .

let e be the electric field defined by e = (18 x, 18 y, 18 z). find the electric flux across m. write the integral over the hemisphere using spherical coordinates, and use the outward pointing normal.

(( a c
))m1 e ·ds = ((
)) f) d? df
b d
where

a = pi/2, b =0 , c =2pi , d =0 ,

using t for ? and p for f,
f) = ? ?

((
))m1 e ·ds =

((
))m2 e ·ds = 0 , so

((
))m e ·ds =

Since is closed, you can use the divergence theorem: The flux of across is

which is 54 times the volume of the hemisphere centered at (0, 0, 0) with radius 1, .

Judging by the question content, you're supposed to find this value by computing the the integral of across and .

Parameterize the hemisphere by

with and . Take the normal vector to to be

The flux of across is

Parameterize the disk by

with and . Take the normal to to be

Then the flux across is

Then the total flux across is , as expected.

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step-by-step explanation: