# 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 ==>
x**F** = 0

x**F** = 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) (d**r**/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**=M**i**+N**j**+P**k**

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

Make a comment using the feedback page

or by sending e-mail to:
<mb49@cornell.edu> or <lcw5@cornell.edu> or
<aps6@cornell.edu>