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We will now be looking at conservative vector fields:
Let’s look at the work formula and apply it to a conservative vector field: Thus we have the Fundamental Theorem of Line Integrals: This should look familiar. Second Fundamental Theorem of Calculus: Thus we have a sort-of-easy way to evaluate line integrals when the vector field is conservative! We would have to do
the work to find the potential function if the field is conservative.
Find which is conservative and earlier in the notes we figured out the potential where the path of integration is C: y=x from (0,0) to (2,2) then y=2x-2 from (2,2) to (5,8) and finally 2 y=x -17 from (5,8) to (1,-16). If this field was not conservative, we would have to parameterize three paths and
evaluate three line integrals to get the answer. But we can use our fundamental theorem of line integrals to evaluate
this – we just need the starting point and the end point. What would happen if we had a closed curve? Our starting point and our ending point would be the same point. If we
have a conservative vector field and we use the fundamental theorem of line integrals – look what happens…. Notice the notation – the circle – it represents a CLOSED curve. Example:
Evaluate the Line Integral: where C is the ellipse with counterclockwise orientation.
You can do a lot of work on this problem to get the final result of zero. OR you may notice that this is a conservative
vector field: Thus for a conservative vector field on a closed curve
Summary Section 14.3
Fundamental Theorem of Line Integrals
Conservative Fields are Independent of Path
where is conservative Section 14.4 Green’s Theorem
Now we will connect a line integral to a double integral with Green’s Theorem. One thing that is important in order to use this theorem is the orientation of C. The bounded region must be to the left
of the orientation. The proof is on page 1157. We use an example to show that it does work and how it can simplify the
calculation of a line integral.
Find the work done by
Unit square with vertices...
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