Calculus Animations,Graphics and Lecture Notes

Flux Integrals and Surface Integrals

Home
About Kelly Liakos
Sponsors
Calculus 1- Limits and Derivatives
Calculus 1 - The Second Derivative
Calculus 1 and 3 Formula Sheets
Chain Rule
Computer Lab Assignments
Conic Sections
Differential Equations
Directional Derivatives/Gradient
Double Integrals
Equillibrium Solutions 1st order DEs
1st order Diff Eqs -Motion
Flux Integrals and Surface Integrals
Infinite Sequences
Infinie Series
Level Curves and Level Surfaces
Line Integrals
Optimization and Related Rates
Optimization for Functions of 2 Variables
Parametric Equations 2-space
Parametric Equations 3-space
Partial Derivatives
Polar Coordinate System
Polar Coordinates- Derivatives and Integrals
PreCalculus
Riemann Sums and the Fundamental Theorem of Calculus
2d order Diff EQS-Motion
2d Partial Derivatives
Supplemental Exercises and Solutions
Tangent Planes/ Differential for f(x,y)
Trigonometry
Trigonometry Applications
Triple Integrals
Unit Tangent Vectors/Unit Normal Vectors
Vectors in General
Vector Valued Functions
Visualizing Limits of Functions of 2 Variables
Work
Links
Special Topics
Scratch Paper
2 Poems

Before beginning you may want to review the notes on vector fields

Notes - Vector Fields

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



The Smart Bunny-A very short story by Kurt Vonnegut Jr.

©2008-2010 Kelly Liakos