15.3 Conservative Vector Fields and Independence of Path

Theorem 15.5 Fundamental Theorem of Line Integrals

Let C be any piecewise smooth curve lying in an open region R and given by . If  is conservative in R and both M and N are continuous on R, then

Ex 1 (A) & (B) Show that the force field is conservative, then  evaluate the line integral over both of the curves:  and .

Since , the field is conservative.

The potential function is

We have two different paths but both start  at (0,0) and end at (8,4). The value of the line integral for each is:

A sketch of the two paths follows. I needed to convert back to non-parametric form to get the graphs.

The whole point was to notice that what ever path we took, the result was the same. I used the theorem to evaluate the integral. Here are both calculated in the usual way:

First path

Second path  

I don't know about you, but I think I have a preferred method! I can't imagine doing this without a CAS! Even using a CAS is way more time consuming that using the Fundamental Theorem of Line Integrals!  Remember, the theorem only holds if the vector field is conservative: It is a gradient field for some potential function.

We need a few definitions to generalize the concept of independence of paths for line integrals.

From the Fundamental Theorem of Line Integrals, as in the example above, if we have a conservative continuous vector valued function F, then the integral 

is independent of the path between any two fixed points in a region R.

Def  Connected

A region is connected if any two points in the region can be joined by a piecewise smooth curve lying entirely in the interior of the region. A drawing follows: 

Theorem 15.6 Independence of Path in Conservative Vector Fields

If F is continuous on an open connected region, then the line integral is independent of path if and only if F is conservative.

Ex 3 For the force field  ,show that the line integral  is independent of path and calculate the work done on an object in moving it along from (0,p/2,1) to (1,p,3).

The integral is independent of path if it is conservative. We can evaluate it by evaluating the potential function at the endpoints of the curve. From what we know of curl of F,

and if this is the zero vector, we have a conservative field. In this case,

Therefore F is conservative. Now we need to find the potential function. 

The potential function must be

The value of the line integral is therefore

How much work is done if an object is moved from one position to another and then returned to the original position? Does it seem logical that since work = force * distance moved, that it must be zero? We need another definition to make sense of the next theorem.

Def Closed Curve

A curve C given by a vector valued function  is closed if .

Theorem 15.7 Equivalent Conditions

Let  have continuous first partial derivatives  in an open connected region R and let C be a piecewise smooth curve on R. Then the following are equivalent:

1. F is conservative (that is F = f for some potential function f.)  

is independent of path.

for every closed curve C in R.

Perhaps a simpler curve can be obtained for the line integral in Ex 2, since ANY path between (0,0) and (8,4) will work.

Ex 1 (C) Evalute the line integral over a simpler path than parts (A) and (B) above: recall one path was , I only show this to remind you that the path connected (0,0) with (8,4). A much simpler path would be . Then the line  integral becomes

This was not nearly so ugly an integration as before!

Conservation of Energy

One of the most important laws of physics is the Law of Conservation of Energy discovered by Michael Farrady. It can be derived via the Fundamental Theorem of Line Integrals (Theorem 15.5). From physics, the kinetic energy of a particle of mass m and speed v is . The potential energy p of a particle at point (x,y,z) in a conservative vector field F is defined as p(x,y,z)= - f(x,y,z) where f is the potential function for F. Consequently the work done by F along smooth curve C from point A to point B is

In other words, the work is equal to the difference in the potential energies of A and B. Suppose then that vector valued function is the position vector for a particle moving along curve C from to . By Newton's Second Law of motion, where F, a, and v' are vector valued functions. The work done by F moving a particle from A to B can be expressed as

Equating the results for W done both ways yields , so with a little algebra we see which implies that the sum of the potential and kinetic energies remains constant from point to point when the field F is conservative.

An example of this law is tossing a ball in the air, at the beginning of the flight the ball has only kinetic energy. At the top of its flight the kinetic is zero and the potential is all the ball has. At the points between, the sum of the kinetic and potential is always the same. (Recall gravitation fields are conservative inverse square fields.)

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Assignment: Read and look at the pictures example 1 in the text pg 1079 and try  page 1086 #3,9,11,15, 25,31,40,41,46