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Unformatted text preview: Math 241 Chapter 15 Dr. Justin O. Wyss-Gallifent 15.1 Vector Fields 1. Define a vector field: Assigns a vector to each point in the plane or in 3-space. 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 2-variable and 3-variable 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....
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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