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Math 201 (section 3)
November 29, 2004
Where we're going from here:
1. Which vector fields are conservative? (i.e., are gradients?)
---Test in terms of dQ/dx-dP/dy
---Path independence for conservative fields
---If F is conservative then integral of F dr around a closed curve is zero
2. More generally, Green's theorem:
integral of F dr on boundary = integral of dQ/dx-dP/dy in region
3. Divergence theorem in two dimensions:
---div(F) = dP/dx + dQ/dy
---This is zero if F represents and incompressible flow
---integral of F-dot-n ds around boundary = integral of div F in region
4. Winding numbers: If F is a continuous, non-zero vector field defined only on the boundary of a region,
when can we extend it to a continuous, non-zero vector field on the entire region?
Dec. 6: Detailed project plans, AND last homework due. We'll fill out course evaluation forms.
Dec. 8: Class party. Class picture. Review. Maybe winding numbers.
Read Sections 17.3 and 17.4.
Problems to turn in:
Section 17.3: 1, 3, 4, 5, 6, 13, 23.
Look at (but don't turn in) 27, 28, 33. On Nov. 24 we saw that the vector field in problem 33
is "sort of" the gradient of what is only "sort of" a function, f(x,y)=atan2(y,x).
Section 17.4: 2, 7, 8.
(end)
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