COALITION MATH 151, FALL 1997

        Group II (S. A. Fulling, assisted by Cary Lasher)

                              Day 10.3


STARTED BY FINISHING THE AREA PROBLEM FROM PREVIOUS DAY.

    1.  Finish the vectors/related rates problem.
STUDENT RESPONSE TO THE CHALLENGE WAS DISAPPOINTING (ESSENTIALLY 
NONEXISTENT).

        A.  Time derivative of distance is not the same as 
        magnitude of velocity vector unless displacement and 
        velocity are parallel (hypotenuse always has same slope).

        B.  If angle between displacement and velocity is small, 
        projection of velocity on displacement is very close to 
        the whole velocity.  (Hope to return to this after Taylor 
        series, since it hinges on cos \theta = 1 - \theta^2/2.)

    2.  Work and line integrals (progression from simple to 
    sophisticated problems)

        A.  One dimension, constant force

        B.  Intermediate case:  step function force

        C.  One dimension, variable force; spring example

        D.  Two dimensions, constant force

            i.  Treat the 2 dimensions separately:  
                F_x \Delta x + F_y \Delta y.

           ii.  Vector treatment:  Parallel component of force
                times displacement.

        E.  Compare the two approaches

            i.  One-dimensional calculus vs. vectors

           ii.  Vector approach sometimes leads to unnecessary 
            calculations (useless square root cancels).

          iii.  Physics and geometry are clearer in the vector 
            approach.
GOT THIS FAR.

        F.  Intermediate case:  Broken line with constant force 
        on each segment.

        G.  Parametrized path with functional force (see next 
        day).