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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!
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!
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.
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!).
Red curve is Kinetic energy K
Blue curve is gravitational potential energy U
Green curve is the Total Energy E = K + U
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”.