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In the first 4 animations we introduce the basic idea of flux. We then present the details in 2 lectures, one on surface integrals in general and one on flux integrals in
particular. Note vector quantities will represented by bold and black lettering.
I'll present the idea of flux in terms of a fluid flow field,however we can also talk about flux in terms of heat flow
or electrical flux, or any number of physical applications. Suppose we have
a surface σ given by z = f(x,y) in 3space and a vector flow field F = f1(x,y)i +f2(x,y)j
+f(x,y)k . Then the Flux is the volume of fluid crossing the surface per unit time. See
Animation 1
Animation 1
To Calculate flux we'll start with the simplest
possible case. Suppose we have a flat surface and the vector field
F is constant in magnitude and direction and is perpindicular to the face of
the surface. See Animation 2
Animation 2
Denote the flow field by v
and the cross-sectional area by A. In time Δt
the amount of Fluid crossing the surface is A||v||Δt . We denote the flux by Φ which is the volume of fluid crossing the surface per unit
time i.e. Φ = (A||v||Δt)/Δt
= A||v||. See
Animation 3 and Diagram 1
Animation 3
Diagram 1
What if the vector field v
is not perpindicular to the surface ? Then we can resolve v
into components parallell and perpindicular to the surface. The component
parallel to the surface does not contribute to the flux. To compute
the perpindicular component we simply project v onto n
the normal to the surface. Then the flux Φ = A ||projnv|| = A v•n See Animation 4 .
Animation 4
We are now in a position to define the flux integral for
a general surface z = f(x.y) and a general vector field v = f1 i + f2 j +f3 k. See the lecture notes below.
Before considering the notes on surface integrals
you may want to first consider the following animation on the local linearity of functions of 2 variables. For a discussion
of local linearity see the page on Tangent Planes and the Differential Also
if you are familiar with surface integrals you can skip to the lecture notes on flux integrals.
Animation-Local Linearity
Notes - Surface Integrals
Find the surface integral of f(x,y,z) = (x2+y2)z
where σ is the portion of the sphere x2 +
y2 +z2 = 4 above the plane z = 1
Solution to Surface Integral Problem
Notes - Flux Integrals
A fluid with density ρ flows with velocity V = y i
+ j + z k . Find the rate of flow of mass upward though the parabaloid z = 9 - 1/4(x2 +
y2) above the x -y plane. |
Solution to Mass Flow Application
Notes - A question involving Stoke's Theorem
Notes - A Second Application of Stokes Theorem
Familiarize yourself with the parametric surface plots
lab on the Computer Lab Assignment page before considering the notes
on Flux over Parameterized Surfaces.
Notes - Flux Integrals over Parameterized Surfaces
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