# Path IndependenceConservative FieldsPotential 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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