Math 294


Lecture 11

Given by Prof. Alan Zehnder on 9/25/95 at 10:10 in Kaufmann Auditorium


Path Independence
Conservative Fields
Potential Functions


In Lecture 10 we worked with Stoke's Theorem. In this lecture, we will show some theorems and examples of path independence, conservative fields, and potential functions.

Theorems:

F=f <==> F is conservative
If F=f, the result (xf)=0 ==> xF = 0
xF = 0 ==>
F is conservative <==>


Path Independence


If is the same for all paths from A to B, then the integral is path independent.
Fields F for which the integral is path independent are conservative

All conservative fields have a potential fuction
The potential function is f where F=f
Example of a potential function:
Gravity - energy = mgy is a potential function
Gravity - force=-d/dy(energy) is the conservative field

Proofs of Theorems

F=f ==> <F conservative
Let C: r(t)=x(t)i+y(t)j+z(t)k, a<=t<=b

df/dt = (df/dx)(dx/dt) + (df/dy)(dy/dt) + (df/dz)(dz/dt) = f (dot) (dr/dt)


where f(B) = f( x(b) , y(b) , z(b) )

Now, we must prove it works the other way:
path independent ==> F=f
Show that there exists f such that M=df/dx, N=df/dy, and P=df/dz where F=Mi+Nj+Pk

Define a point A where f(A) = 0
Define B, f(B) = (We can only say this because F is path independent.




This proof can be extended to incude N and P also if we choose different B0.


Lecture written by Aric Shafran on Thursday, October 6, 1995
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