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 quite well and (3) what you want more information on. 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 large as h->0. But, if we can tease out more information, 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 .