MA 242.003
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Transcript of MA 242.003
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MA 242.003
• Day 53 – April 2, 2013• Section 13.2: Finish Line Integrals• Begin 13.3: The fundamental theorem for line
integrals
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Extension to 3-dimensional space
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A major application: Line integral of a vector field along C
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A major application: Line integral of a vector field along C
We generalize to a variable force acting on a particle following a curve C in 3-space.
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Line Integrals with respect to x, y and z.
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Line Integrals with respect to x, y and z.
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Line Integrals with respect to x, y and z.
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Line Integrals with respect to x, y and z.
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Line Integrals with respect to x, y and z.
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Line Integrals with respect to x, y and z.
This says any line integral with respect to x, y and/or z can be REWRITTEN as a line integral of a vector field.
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Line Integrals with respect to x, y and z.
This says any line integral with respect to x, y and/or z can be REWRITTEN as a line integral of a vector field.For example, the line integral
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Line Integrals with respect to x, y and z.
This says any line integral with respect to x, y and/or z can be REWRITTEN as a line integral of a vector field.For example, the line integral
the line integral of the vector field
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Result: Any line integral of the form
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Result: Any line integral of the form
can be reformulated as a line integral of a vector field
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Section 13.3The Fundamental Theorem for Line Integrals
In which we characterize conservative vector fields
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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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Before we prove this theorem I want to recall a result from the section on the chain rule:
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Before we prove this theorem I want to recall a result from the section on the chain rule:
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Before we prove this theorem I want to recall a result from the section on the chain rule:
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Proof:
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(continuation of proof)
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See your textbook for the proof!
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See your textbook for the proof!
Note that we now have one characterization of conservative vector fields on 3-space.
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See your textbook for the proof!
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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See your textbook for the proof!
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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Recall the following theorem from chapter 12:
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Recall the following theorem from chapter 12:
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Recall the following theorem from chapter 12:
Let’s use this property to investigate the properties of components of a conservative vector field.
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Recall the following theorem from chapter 12:
Let’s use this property to investigate the properties of components of a conservative vector field.
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(The calculation)
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Hence we have proved:
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Hence we have proved:
This is another characterization of conservative vector fields!
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Hence we have proved:
This is another characterization of conservative vector fields!
The question arises:
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Hence we have proved:
This is another characterization of conservative vector fields!
The question arises: Is the CONVERSE true?
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Hence we have 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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