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Intermediate CalculusandLinear Algebra
Jerry L. KazdanHarvard UniversityLecture Notes, 1964–1965
i
Preface
These notes will contain most of the material covered in class, and be distributed beforeeach lecture (hopefully). Since the course is an experimental one and the notes writtenbefore the lectures are delivered, there will inevitably be some sloppiness, disorganization,and even egregious blunders—not to mention the issue of clarity in exposition. But we willtry. Part of your task is, in fact, to catch and point out these rough spots. In mathematics,proofs are not dogma given by authority; rather a proof is a way of convincing one of thevalidity of a statement. If, after a reasonable attempt, you are not convinced, complainloudly.Our subject matter is intermediate calculus and linear algebra. We shall develop thematerial of linear algebra and use it as setting for the relevant material of intermediatecalculus. The ﬁrst portion of our work—Chapter 1 on inﬁnite series—more properly belongsin the ﬁrst year, but is relegated to the second year by circumstance. Presumably this topicwill eventually take its more proper place in the ﬁrst year.Our course will have a tendency to swallow whole two other more advanced courses,and consequently, like the duck in Peter and the Wolf, remain undigested until regurgitatedalive and kicking. To mitigate—if not avoid—this problem, we shall often take pains tostate a theorem clearly and then either prove only some special case, or oﬀer no proof atall. This will be true especially if the proof involves technical details which do not helpilluminate the landscape. More often than not, when we only prove a special case, theproof in the general case is essentially identical—the equations only becoming larger.September 1964
ii
Afterward
I have now taught from these notes for two years. No attempt has been made to revisethem, although a major revision would be needed to bring them even vaguely in line withwhat I now believe is the “right” way to do things. And too, the last several chaptersremain unwritten. Because the notes were written as a ﬁrst draft under panic pressure,they contain many incompletely thought-out ideas and expose the whimsy of my passingmoods.It is with this—and the novelty of the material at the sophomore level—in mind, thatthe following suggestions and students’ reactions are listed. There are three categories,A), Material that turned out to be too diﬃcult (they found rigor hard, but not many of the abstractions), B), changes in the order of covering the stuﬀ, and C), material—mainlysupplementary at this level—which is not too hard, but should be omitted if one ever hopesto complete the ”standard” topics within the conﬁnes of a year course.(A)
It was too hard
(unless one took vast chunks of time).(1) Completeness of reals. Only “monotone sequences converge” is needed for inﬁniteseries.(2) Term-by-term diﬀerentiation and integration of power series. The statement of the main theorem should be fully intelligible—but the proof is too complicated.(3) Cosets. This is apparently too abstract. It might be possible to do after ﬁndinggeneral solutions of linear inhomogeneous O.D.E.’s.(4)
L
2
and uniform convergence of Fourier series. Again, all I ended up doing was totry to state what the issues were, and not to attempt the proof. The ambitiousstudent should be warned that my proof of the Weierstrass theorem is opaque(one should explicitly introduce the idea of an approximate identity).(5) Fundamental Theorem of Algebra. The students simply don’t believe inequalitiesin such profusion.(6) I you want to see rank confusion, try to teach the class how to compute higherorder partial derivatives using the chain rule. That computation should be oneof the headaches of advanced calculus.(7) Existence of a determinant function. I don’t know a simple proof except for theone involving permutations—and I hate that one.(8) Dual spaces. As lovely as the ideas are, this topic is too abstract, and to myknowledge, unneeded at this level where almost all of the spaces are either ﬁnitedimensional or Hilbert spaces. One should, however, mention the words “vector”and “covector” to distinguish column from row vectors. I forgot to do so in thesenotes and it did cause some confusion.
iii(B)
Changes in Order and Timing
. The structure of the notes is to investigate barelinear spaces, then linear mappings between them, and ﬁnally non-linear mappingsbetween them. It is with this in mind that
linear
O.D.E.’s came before
nonlinear
mapsfrom
R
n
→
R
. The course ended by treating the simplest problem in the calculusof variations as an example of a nonlinear map from an inﬁnite dimensional spaceto the reals. My current feeling is to consider linear
and
non-linear maps between
ﬁnite
dimensional spaces before doing the inﬁnite dimensional example of diﬀerentialequations.The ﬁrst semester should get up to the generalities on solving
LX
=
Y
, p. 319[incidentally, the material on inverses (p. 355 ﬀ) belongs around p. 319]. Moststudents ﬁnd the material on linear dependence diﬃcult—probably for two reasons:1) they are not used to formal deﬁnitions, and ii) they think they have learned atechnique for doing something, not just a naked deﬁnition, and can’t quite ﬁgure out just what they can do with it. In other words, they should feel these deﬁnitions aboutthe anatomy of linear spaces are similar to those describing a football ﬁeld and of little value until the game begins—i.e., until the operators between spaces make theirgrand entrance.Because of time shortages, the sections on linear maps from
R
1
→
R
n
and
R
n
→
R
1
,pp. 320-41 were regrettably omitted both years I taught the course. The notes werewritten so that these sections can be skipped.(C)
Supplementary Material
. A remarkable number of fascinating and important topicscould have been included—if there were only enough time. For example:(1) Change of bases for linear transformations (including the spectral theorem).(2) Elementary diﬀerential geometry of curves and surfaces.(3) Inverse and implicit function theorems. These should be stated as natural gener-alizations of the problems of a) inverting a linear map, b) ﬁnding the null spaceof a linear map, and c) generalizing dim
D
(
L
) = dim
R
(
L
) + dim
N
(
L
) all tolocal properties of nonlinear maps via the tangent map.(4) Change of variable in multiple integration. Determinants were deliberately in-troduced as oriented volume to make the result obvious for linear maps andplausible for nonlinear maps.(5) Constrained extrema using Lagrange multipliers.(6) Line and surface integrals along with the theorems of Gauss, Green, and Stokes.The formal development of diﬀerential forms takes too much time to do here.Perhaps a satisfactory solution is to restrict oneself to line integrals and thesetheorems in the plane, where the topological diﬃculties are minimal.(7) Elementary Morse Theory. One can prove the Morse inequalities easily for thereal line, the circle, the plane, and
S
2
merely by gradually ﬂooding these setsand observing the number of lakes and shore line changes only at the criticalpoints.

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