COALITION MATH 151, FALL 1997

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

                              Day 13.1


Double integrals and Green's theorem (postponed from previous 
week)

    1.  Quick remarks on definition (partition of a rectangle 
    into small rectangles to produce a Riemann sum).

    2.  Interpretations and applications

        A.  Volume (of a certain type of 3-dim region)

        B.  Total mass (or cost, etc) of a plate with varying 
        density.  (Mass of a 3-dim object will require a triple 
        integral.)  In particular, if the density is 1, the 
        double integral is the area of the 2-dim region; this is 
        of interest when the region is more complicated than a 
        rectangle.

        C.  Flux of a vector field through a surface (brief 
        mention).  This uses the perpendicular part of the field 
        as the line integral uses the parallel part.

        D.  Green's theorem -- the main reason for bringing up 
        double integrals at this point:

    3.  Green's theorem:  Evaluate a line integral around a 
        closed path (simple, counterclockwise, no singularities 
        of the field inside).  Call the interior region D.

        A.  We know that the integral is zero if 

            d{F_y}/dx - d{F_x}/dy = 0.

        B.  Generalization:  The line integral equals the double 
        integral over D of d{F_y}/dx - d{F_x}/dy.  (Next 
        semester, when we study the vector cross product, it will 
        be explained why your instructor gives himself the 
        "Gig'em Aggies" sign to remember which term gets the 
        minus sign.)

        C.  Special case:  Find a vector field F for which 
        d{F_y}/dx - d{F_x}/dy = 1.  (There is more than one such 
        F.)  Then the line integral equals the area of the 
        region!  This enables you to complete your lab 
        assignment on the area of Texas.

    4.  Evaluation of double integrals as iterated integrals.  
    (This will continue through the next day.)

        A.  Think of slicing the 2-dim region into strips.  This 
        is just like finding areas, except generalized to 
        integrands not equal to 1 (yielding mass, volume, etc.).

        B.  Show rectangular example.

        C.  Activity:  Another rectangular example, Stewart 
        13.2.15 (ed. 3).  Do both ways. 
GOT THIS FAR ON 13.1, LEAVING THE PRESENTATION TO DAY 13.2.  THE 
CLASS AND I AGREED THAT VERTICAL SLICING SUFFICED, AND WE 
PROCEEDED TO THE NEXT ACTIVITY.

        D.  Activity:  A nonrectangular case, Stewart 13.3.7.
        Individual brainstorming followed by team solution.  Hope 
        to get both horizontal and vertical solutions.
THIS EXAMPLE TURNS OUT TO BE TOO SYMMETRICAL FOR THE HORIZONTAL 
SOLUTION TO BE INSTRUCTIVE.  STUDENTS FOUND THE VERTICAL 
CALCULATION EASY.

        E.  Activity:  A demonstration of Green's theorem, 
        Stewart 14.4.3.  Every team should do both ways. 
LEFT PRESENTATION FOR DAY 14.1.