COALITION MATH 151, FALL 1997

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

                              Day 12.1


(ORIGINALLY PLANNED FOR DAY 11.3)  
BEGAN BY FINISHING SCRIPT FOR DAY 11.2.

    1.  Gradients, potential energy, and all that

        A.  I shall convince you that you already know how to 
        work with partial derivatives.  Call on a team:
TO SAVE TIME I WROTE THIS OUT MYSELF (IN ADVANCE) INSTEAD OF 
CALLING ON A TEAM.

         i.  Find derivative of \sqrt{x^2 - 5} with respect to x. 

        ii.  Find derivative of \sqrt{x^2 - C} with respect to x, 
        if C is an arbitrary constant. 

       iii.  Find derivative of \sqrt{x^2 - y} with respect to x.

        iv.  Find derivative of \sqrt{x^2 - y} with respect to y.

        Don't worry now about why the \partial notation is used.

        B.  Unlike implicit differentiation, now we differentiate 
        with respect to x assuming y is fixed.  Earlier y and x 
        were related (linked by an equation).  Now x and y are 
        two independent variables.

        C.  Fact:  For any "normal" f, the order of mixed 
        partials doesn't matter.  (Verify for our example above.)

        D.  Define gradient of f(x,y)

        E.  After these preliminaries we finally get to the 
        point.  Suppose we have a vector field F(x,y) for which 
        the line integrals (work) are always independent of path.
        Define g(\vec r) as the line integral to \vec r from the 
        origin (or any fixed point).  Fact:  F = grad g.
        (In previous day's example:  g = x^2 sin y.)

        F.  Fundamental theorem for line integrals:  Given a 
        scalar function g, we have

        \int_{r_0}^{r_1} \nabla g . d\vec{r} = g(r_1) - g(r_2).

        G.  Define V = - g = potential energy corresponding to F.  
        Then work-energy theorem plus fundamental theorem =>

        m|dr/dt|^2/2 + V(r) = constant (indep. of time) 
           = total energy. 

        Under these conditions F is called "conservative".

    2.  Summary:  For \int_{r_0}^{r_1} F.dr, the following are 
    equivalent (you get all or none):

        A.  The integral can be evaluated by a vectorial version 
        of the fundamental theorem of calculus.

        B.  There exists a scalar function V such that 
    
            F = - grad V.

        C.  The integral is independent of the path followed from 
        r_0 to r_1.

        D.  The line integral around any closed path (r_1 = r_0) 
        is zero.

        E.  dF_x/dy = dF_y/dx everywhere.

        F.  Energy is conserved, if F is interpreted as a force.
        (In Sophomore Coalition language:  The energy 
        conservation accounting principle does not involve any 
        nonzero inflow or outflow terms.)