This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 241 Chapter 15 Dr. Justin O. WyssGallifent 15.1 Vector Fields 1. Define a vector field: Assigns a vector to each point in the plane or in 3space. Can be visualized as loads of arrows. Can represent a force field or fluid flow  both are useful. 2. Two important definitions. Often before I do these I define = x + y + k so that gradient, divergence and curl all make sense with how is used. (a) The divergence F = M x + N y + P z gives the net fluid flow in/out of a point (very small ball). (b) The curl F gives the axis of rotation of the fluid at a point. 3. For a function f we saw the gradient f is a VF. In fact its a special kind of VF. Any VF which is the gradient of a function f is conservative and the f is a potential function . There are two facts to note: (a) If F is conservative then F = 0 and consequently if F negationslash = 0 then F is not conservative. Moreover if F = 0 and F is defined for all ( x,y,z ) then F is conservative. (b) If we have F we can tell if its conservative by the above method and we can find the potential function too using the iterative method. Make sure to do 2variable and 3variable cases. 15.2 Line Integrals (of Functions and of VFs) 1. If C is a curve and f gives the density at any point then we can define the line integral of f over/on C , denoted integraltext C f ds , as the total mass of C . We evaluate it by parametrizing C as r ( t ) on [ a,b ] and then integraltext C f ds = integraltext b a f ( x ( t ) ,y ( t ) ,z ( t ))  r ( t )  dt . The result is independent of the parametrization and the orientation....
View
Full
Document
This note was uploaded on 02/27/2012 for the course MATH 2 taught by Professor Staff during the Spring '08 term at Maryland.
 Spring '08
 staff
 Math

Click to edit the document details