30aug02(F1)

Week 1 had only two days, Wednesday and Friday.
We will refer to these as W1 and F1. Thus there
is no M1, nor M2 because next Monday is Labor Day.
In preparation for the next class, W2, please read 
sections 1.5 and read into 1.6.

The principal topics were the definition of the
derivative of a function f at x,  as that object
f'(x) which fits into the formula
 
f(x+h) = f(x) + f'(x)h + eh, where lim e(x,h)=0 
                                   h->0
[In class, I used \theta and \phi for the vanishing
error term, here denoted by e.]

This approach enables us to calculate the chainrule
instead of proving the truth of an apriori formula.

Defining H=fog as H(x)=f(g(x)) we substitute
H(x+h)=f(g(x+h))=f(g(x)+g'(x)h + eh).
Observe how, just as h displaces x to x+h, the 
corresponding displacement of g(x) is (g'(x)+e)h. 
Continuing
      =f(g(x)) + f'(g(x))(g'(x)+e)h + E(g'(x)+e)h
where E is the vanishing error for f' at x. 
Continuing
      =H(x) + f'(g(x))g'(x)h + (vanishing error)h
which demonstrates that f'(g(x))g'(x)h plays the 
role of H'(x). 

On Friday I showed the math majors how to make the 
error term 
      f'(g(x))eh + E(g'(x)+e) 
vanish with h at x. The required lemma is to show
that if E vanishes at g(x) with a displacement 
which vanishes at x with h, also vanishes at x with h.

Students allergic to this sort of analysis should 
ignore it. We can do without it in this course. 

The historical lesson associated with this definition
of the difference is the enormous utility of Leibniz'
notation for the differential calculus, such as

If u=f(y) then du=f'(y)dy, so that if y=g(x), and
therefore dy=g'(x)dx, then du=f'(y)g'(x)dx. 

The next step is to generalize the interpretation of
the derivative to the multivariate context. Thus 
for f:R^p -> R^n, consider the various cases for (p,n)

case(1,3): f is a parametrized curve (= path) and 
f'(t) is the velocity vector of the path at t.  

case(3,3): eg1 f is a change of coordinates, 
but also eg2 f is a vector field 

case(3,1): f is a scalar field, it assignes a number to 
each point. If u=f(x,y,z), then, by the (multivariate) 
chain-rule:  du=f_x dx + f_y dy + f_z dz,
where f_x denotes the partial derivative of f wrt x, etc.

Note that this expression is a dotproduct between the 
gradient vector and the displacement vector.  

case(2,3): f parametrizes a surface.

case(2,1): We will consider  function f:M^2->R which are 
defined only on a surface, and 

case(1,2):  f:R^1->M^2, a path (parameterized curve) 
on a surface.