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

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.