Week 1, 20 and 22 January 1999

Assignment for 22jan99: Read "into" Chapter 2 of Kaplan and
be prepared to say (1) how far you got (2) what you understood

Here is a slightly expanded version of what I said in class.

Please note that .html does provide the usual calculus symbols,
and decorations of letters. So we will establish this convention:

x_i means "x sub i"  and
x^2 mans  "x super 2" i.e. "x-squared"

When it is necessary to write a fat letter, which would be written
"x underlined" I will write "_x" .
But most of the time we can leave off that decoration.

Thus "_x = x_1 x_2 x_3"  says that "dim _x = 3.

The important augmentation to Kaplan and to what you may have learned
in Math 242/243/245 is that we express limits in terms of
"null functions" also called "error functions" or "vanishing functions."
Since with don't have a \theta in .html I will use Q(h) when I mean that

(0) lim  Q(h) = 0 .
h->0

Thus

(1)  f(x+h)= f(x) + Q(h)
says that f is continuous at h.

In general Q is a function of f,x, and h. If we can factor out an
h in the sense that Q(x,h)=K(x,h)h,

(2) f(x+h) = f(x) + K(x,h)h

then continuity requires only that K(x,h) does not get infinitely
namely that K(x,h) = f'(x) + Q'(x,h), where f' is an expression that
depends ONLY on x (not also on h) and Q' is another null function,
then f is differentiable at x and its derivative is f'(x).

(3) f(x+h) = f(x) + (f'(x) + Q')h = f(x) + f'(x)h + Q'(x,h)h .
d
This is a good definition of the derivative, because it makes sense
when dim x = n > 1. For then dim h = n in order to be able to add
these quantities, and Q,K,Q' and f'(x) are all matrices of the same
dimensions p rows of n columns, where p = dim(f). It is a theorem
that the entries of f'(x) are the partial derivatives of the componenents
(4) u = f(x) , f'_ij(x) = du_i/dx_j
where we (obviously) mean \partial for d .