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Derivatives in N-Dimensional Spaces

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Derivatives in N-Dimensional Spaces 5 DERIVATIVES IN N-DIMENSIONAL VECTOR SPACES A Introduction , -' ,.- * -..: The approach of our textbook (as d&ioped in the previous two units) in emphasizing the geometrical aspects of partial derivatives, directional derivatives, and the gradient is, in many ways, an excellent approach. We get a picture of what appears to be happening, and, especially for the beginning student, this is much more meaningful than a more precise and general discussion that would apply to all vector spaces, regardless of dimension. In fact, if we want to use the latter approach, it is wise to use the former first. This is in accord with many such decisions which have already been made .in our previous treatment of mathematics. As an example, within the framework of our present course, notice how carefully we ex- ploited the geometric aspects of vectors as arrows before we even mentjoned n-dimensional vector spaces. And, in this respect, recall how sophisticated many of our "arrow" results were in their own right. That is,.granted that arrows were simpler than n-tuples, when all we had were arrows, there were still some rather difficult ideas for us to assimilate. In any event, our aim in this chapter is to continue within the spirit of the game of mathematics which, thus far, has served us so well. Namely, we have introduced n-dimensional vector spaces so that we could utilize the similarity in form between the previously studied functions of the form f(x) and our new functions of the form f(x). For example, when it came time to define continuity and limits for functions of several real variables, we rewrote f(xl, ...,xn) as f (5) and then mimicked the f (x) situation by copying definitions verbatim, subject only to making the appropriate "vectorizations." Rather than review this procedure, let us instead move to the next plateau. Using the approach developed in Chapter 4 of these supplementary notes, once we had defined what limits and continuity meant for any function, f :E"+E, our next logical step would have been to mimic our definition of f' (a) , and thus "induce" a meaning of f ' (5) where ae~n. I . ' . , Our aim in this chapter is to see where such an attempt would have led us and, once this is done, to see how our "new" concept of differentiation is related to the more traditional concept of differentiation as it was presented in the previous two units. B A New Type of Quotient From the definition £1 1.1 = 1"a+~x)-fAX (a) 1 it seems natural that if now f:En+E and -afzEn, we should define f' (a)- where (2) is obtained from (1) by the appropriate "vectorizations. " In arriving at (21, the most subtle point is to observe that we must replace Ax by Ax,- since, among other things, -a+Ax has no meaning (i.e., we have no rule for adding a vector [a]- to a real number [AX]). Once Ax is replace by Ax,- it is clear that 0 must be replaced by -0. If we next look at the bracketed expression in (2), we get a rather elegant insight as to how new mathematical concepts are often born. For, while at first glance, the derivation of (2) from (1)may seem harmless, a second glance shows us that we have "invented" an operation which we have never encountered before in our study of vectors (either as arrows or as n-tuples). More specifically, in the bracketed expression, our numerator is a real number (i.e., while -a is a vector [n-tuple] , recall that our notation f (5) indicates that our "output" is a number), and our denominator is a vector! * Thus, whether we like it or not, if we decide to accept (2) as a C starting point, we are obliged to investigate the meaning of - where c denotes an arbitrary but fixed real number and -v an arbitrary but fixed vector. Notice, of course, that we could elect to abandon (2), but, in the usual spirit of things, it hardly seems wise to abandon an approach that has been fruitful simply because one potential problem arrises. Rather, it seems wiser for us to try, at least, to find a practical definition of what it means to divide a number by a vector. Once we have agreed on this course of action, we again fall back on our previous experience and recall how we defined division in the case of two numbers. We defined to be that number which when multiplied by b yielded a. In terms of a more computational form, a was defined by *. l . 9 b "times" ;= a With this in mind, it seems that a very natural way to mimic (3) is C to define -v by - , .. ,* ;:i "1 . ". C .\ 4 ... v "tipest' - = c .., '* s - -v (4) This immediately suggests that unless we want to invent new forms of multiplication, (and nothing precludes this possibility, except that we already have enough problems1 the "times" in (4) must denote the dot product, for according to (4) we must multiply a vector (v)- by "something" to obtain a number, and of the types of multiplication we have discussed, only the dot product allows us to multiply a vector by "something" (in fact, by another vector) to obtain a number. Thus, the "times" in (4) must denote a dot product. And, it therefore C appears that v- must, itself, be a vector, since, as we have just said, the dot-product combines two vectors to produce a number. There is, however, a little complication that presents itself here. The problem actually existed when we talked about the quotient of two numbers, but, except in the rather special case in which the denominator was zero, the problem never occurred. The point we are driving at is that when we say ";is +he number which when multiplied by b yields a," what gives us the right to say "the?"- Why can't there be more than one such number, or for that matter, why not no such number? The language of sets provides us with a nice explanation. Suppose.: . we define 2 to be the -set of all numbers, c, such that hxc = a. That b - - is, -.. - , c ~. Then, unless b = 0'. the -set a consists of the single 'number a* 6 (e.g., = {c:3c=6) = {6;3) = (211, and hence there is no harm in confusing the set with the number E, where in the latter context, is used as the fraction which denotes a*. Mimicking (5) we may also define 2 to be a set; namely V C The problem is that, unless 1= g and c # 0 (in which case 2 = $), the many - set -v has infinitely elements! Before we document this last remark, let us make sure that it is clear to you that the set described in (6) is a well-defined set regardless of the dimension of the vector space. In other words, recall that in the last chapter of these supplementary notes, we generalized the definition of a dot product so that it existed in any dimensional space. By way of a brief review, if x = (xl, ...,xn and y = (yl ,...,yn) then -x-y = xlyl+ ...+ xn y n' 4 Thus, by way of an example, suppose c = 5 and -v = (1,2,3,4)~E. Then by (61, * If b-0, we have seen that if a#O then :=I since no number times 0 can yield a non-zero number, and if b=O and a=O, we have seen that a 0 (i.e., ,) is the set of numbers, since any number times zero is zero. Notice that while it may be difficult to think of pictorially, (1,2,3,4 ) (7) shows us that it is a "very real" set, at least in the sense that it is the solution set of the "very real* linear equation in four unknowns In fact, it might seem more natural to you if we restated this last remark in sort of a 'reverse" way. Suppose in a traditional math course we were asked to find all solutions of the equation given by (8). Among other things, we may pick values for x2, x3, and x4 completely at random, and once these random choices are made, xl can then be uniquely determined from (8) simply by letting x1=5- 2x2 - 3x3 - 4x4. In modern language, this means that the solution set of (8) has infinitely many elements, and, by (7), another name for this infinite solution set is 5 (1,2,3,4) In still other words, while we may not be used to thinking of the solutions of an equation like (8) as being points in a 4-dimensional vector space, the fact is that, conceptually, the idea is sound. We admit, however, that in the spirit of the text, there is probably more satisfaction if we think of the special cases in which we may view our vectors (n-tuples) as arrows and see what the geometric * implications are. For the sake of a bit of simplicity (and we shall show in a moment that this restriction in no way loses any generality), let us assume that in (6), denotes a unit vector (because if -v is a unit vector, x*~is simply the projection of -x in the direction of -v while if v is not a unit vector, we must merely come to grips with the more computational fact that -x0v- is a scalar multiple of ', S- the projection of 5 in the direction of x).
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