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.
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
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.