COALITION MATH 151, FALL 1997

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

                              Day 6.3


    1.  Unfinished business from previous day

        A.  Comments on tests and grades

        B.  Part iii of parametric circle exercise

    2.  A scenario to motivate vector projections and their 
    relation to coordinate systems.  (Eventually this should be 
    added to the Web page for day 6.1.) 

        A.  Suppose you work for an electric company with a power 
        line stretching across the county along a path such as 
        y = 3x - 5.  The EPA complains that power line is killing 
        birds; effect is more extreme near the power plant, 
        because of voltage drop. 

        B.  Will you formulate the physics of this problem in 
        terms of Cartesian coordinates of points on the wire, and 
        of bird positions?  The sensible thing is to work with 
        the relevant variables, distance along wire (making the 
        science of the wire and birds in contact with it 
        effectively one-dimensional) and perpendicular distance 
        from wire.  (Overlay coordinate grid on transparency.)

        C.  If m and n are unit vectors along these axes, then 
        the new coordinates are c_1 and c_2 , with 
    
        \vec r = c_1 m + c_2 n .

        D.  Because of orthonormality conditions on m and n, one 
        obtains the c_j by dot product of \vec r with m and n.
        Thus c_1 is the scalar projection of \vec r onto m, and so 
        on.

        E.  Repeat the discussion with the standard Cartesian 
        coordinates and the unit vectors i and j:

        \vec r = (x,y) = x i + y j , where x = i.\vec r, etc.

    3.  Polar coordinates and polar unit vectors  (Summarize the 
    Web page and show the diagrams as transparencies.)

        A.  Definition of polar coordinates (the classic spider 
        web)

        B.  Definition of polar unit vectors  (Overlay.)

        C.  Run through the two calculations of the unit vectors 
        presented on Web page.

        D.  Technicalities (nonuniqueness) in the definition of 
        theta 
ALL OF ABOVE WAS DONE EXCEPT 3D, WHICH MAY BE RETURNED TO LATER.