MA 242.003 Day 55 – April 4, 2013 Section 13.3: The fundamental theorem for line integrals –...
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Transcript of MA 242.003 Day 55 – April 4, 2013 Section 13.3: The fundamental theorem for line integrals –...
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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
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Section 13.3The Fundamental Theorem for Line Integrals
In which we characterize conservative vector fields
And generalize the FTC formula
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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.
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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!
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We proved:
This is another characterization of conservative vector fields!
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We proved:
This is another characterization of conservative vector fields!
The question arises: Is the CONVERSE true?
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We proved:
This is another characterization of conservative vector fields!
The question arises: Is the CONVERSE true? YES!
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Proof given after we study Stokes’ theorem in section 13.7.
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FINDING POTENTIAL FUNCTIONS
QUESTION: Now that we have a conservative vector field,
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FINDING POTENTIAL FUNCTIONS
QUESTION: Now that we have a conservative vector field, how do we find potential functions?
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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.
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Illustration of the method:
F = < 2x + z , 2y + z, 2z + x + y>
Conservative?
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Illustration of the method:
F = < 2x + z , 2y + z, 2z + x + y>
Find potential functions:
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(continuation of example)
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You can construct your own “find the potential functions” as follows:
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You can construct your own “find the potential functions” as follows:
1. Choose a function f(x,y,z) . For example:
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You can construct your own “find the potential functions” as follows:
1. Choose a function f(x,y,z) . For example:
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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:
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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!).
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An Application: The Law of Conservation of Total Energy
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An Application: The Law of Conservation of Total Energy
t=at=b
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t=at=b
We calculate the work done
in two different ways.
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t=at=b
We calculate the work done
in two different ways.
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t=at=b
We calculate the work done
in two different ways.
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An Identity: We can derive a very useful identity by differentiating the function
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t=at=b
We calculate the work done
in two different ways.
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t=at=b
We calculate the work done
in two different ways.
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t=at=b
We calculate the work done
in two different ways.
![Page 38: 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.](https://reader033.fdocuments.us/reader033/viewer/2022051517/56649ecb5503460f94bd9903/html5/thumbnails/38.jpg)
t=at=b
We calculate the work done
in two different ways.
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Red curve is Kinetic energy K
Blue curve is gravitational potential energy U
Green curve is the Total Energy E = K + U
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D open Means does not contain its boundary:
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D open Means does not contain its boundary:
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D simply-connected means that each closed curve in D contains only points in D.
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D simply-connected means that each closed curve in D contains only points in D.
Simply connected regions “contain no holes”.
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D simply-connected means that each closed curve in D contains only points in D.
Simply connected regions “contain no holes”.
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