The Surface Integral for Flux
The flux of a three-dimensional vector field F a cross an oriented surface
S in the direction of n is given by the formula:
Where
and
Example:
outward through the surface S cut from Find the flux
by the planes x = 0 and the semicircular cylinder
x = 1.
Solution:
The outward normal to S may be calculated from the gradient of
to be
we also have With
The value of F.n on the surface is given by the formula
on S
Therefore, the flux of F outward through S is:
Masses and Moments of thin shells
(1) Mass:
is the density at (x, y, z) , where
(2) First moments about the coordinate planes:
(3) Coordinates of center of mass:
(4) Second moments (moments of inertia):
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