MA 242.003

• Day 55 – April 4, 2013• Section 13.3: The fundamental theorem for line

integrals– Review theorems– Finding Potential functions– The Law of Conservation of Total Energy

Section 13.3The Fundamental Theorem for Line Integrals

In which we characterize conservative vector fields

And generalize the FTC formula

Note that we now have one characterization of conservative vector fields on 3-space. They are the only vector fields whose line integrals are independent of path.

Note that we now have one characterization of conservative vector fields on 3-space. They are the only vector fields whose line integrals are independent of path.

Unfortunately, this characterization is not very practical!

We proved:

This is another characterization of conservative vector fields!

We proved:

This is another characterization of conservative vector fields!

The question arises: Is the CONVERSE true?

We proved:

This is another characterization of conservative vector fields!

The question arises: Is the CONVERSE true? YES!

Proof given after we study Stokes’ theorem in section 13.7.

FINDING POTENTIAL FUNCTIONS

QUESTION: Now that we have a conservative vector field,

FINDING POTENTIAL FUNCTIONS

QUESTION: Now that we have a conservative vector field, how do we find potential functions?

FINDING POTENTIAL FUNCTIONS

QUESTION: Now that we have a conservative vector field, how do we find potential functions?

SOLUTION: Integrate the three equations,

one at a time, to find the potentials for F.

Illustration of the method:

F = < 2x + z , 2y + z, 2z + x + y>

Conservative?

Illustration of the method:

F = < 2x + z , 2y + z, 2z + x + y>

Find potential functions:

(continuation of example)

You can construct your own “find the potential functions” as follows:

You can construct your own “find the potential functions” as follows:

1. Choose a function f(x,y,z) . For example:

You can construct your own “find the potential functions” as follows:

1. Choose a function f(x,y,z) . For example:

You can construct your own “find the potential functions” as follows:

1. Choose a function f(x,y,z) . For example:

2. Then compute its gradient:

You can construct your own “find the potential functions” as follows:

1. Choose a function f(x,y,z) . For example:

2. Then compute its gradient:

3. Now you have a conservative vector field –

so find its potential functions (you already know the answer!).

An Application: The Law of Conservation of Total Energy

An Application: The Law of Conservation of Total Energy

t=at=b

t=at=b

We calculate the work done

in two different ways.

t=at=b

We calculate the work done

in two different ways.

t=at=b

We calculate the work done

in two different ways.

An Identity: We can derive a very useful identity by differentiating the function

t=at=b

We calculate the work done

in two different ways.

t=at=b

We calculate the work done

in two different ways.

t=at=b

We calculate the work done

in two different ways.

t=at=b

We calculate the work done

in two different ways.

Red curve is Kinetic energy K

Blue curve is gravitational potential energy U

Green curve is the Total Energy E = K + U

D open Means does not contain its boundary:

D open Means does not contain its boundary:

D simply-connected means that each closed curve in D contains only points in D.

D simply-connected means that each closed curve in D contains only points in D.

Simply connected regions “contain no holes”.

D simply-connected means that each closed curve in D contains only points in D.

Simply connected regions “contain no holes”.