Advanced Multivariable Differential Calculus

1 Advanced Multivariable Differential CalculusJoseph BreenLast updated: December 24, 2020 Department of MathematicsUniversity of California, Los AngelesContentsPreface31 Preliminaries42 Euclidean elements of Euclidean space .. algebra of Euclidean space .. geometry of Euclidean space .. products .. important inequalities ..143 Some of sequences .. of functions ..254 Some Linear maps .. multiplication .. standard matrix of a linear map .. multiplication ..405 Curves, Lines, and curves .. cross product ..496 the derivative .. the derivative in one variable .. Multivariable derivative .. of the derivative .. derivatives and the Jacobian .. the chain rule .. order derivatives .. values and optimization.

2 Extrema on closed and bounded domains .. of the gradient .. derivatives .. curves and contour plots .. planes ..917 The Inverse and Implicit Function inverse function theorem .. implicit function theorem .. method of Lagrange multipliers .. of the Lagrange multiplier method .. multipliers with multiple constraints ..122A More Linear Algebra126B Proof of the Inverse and Implicit Function Some preliminary results .. contraction mapping theorem .. mean value inequality .. Proof of the inverse function theorem .. Proof of the implicit function theorem .. The constant rank theorem ..1292 PrefaceThese notes are based on lectures from Math 32AH, an honors Multivariable differentialcalculus course at UCLA I taught in the fall of 2020.

3 Briefly, the goal of these notes is todevelop the theory of differentiation in arbitrary dimensions with more mathematical ma-turity than a typical Calculus class, with an eye towards more Advanced math. I wouldn tgo so far as to call this amultivariable analysistext, but the level of rigor is fairly high. Thesenotes borrow a fair amount in terms of overlying structure fromCalculus and Analysis inEuclidean Spaceby Jerry Shurman, which was the official recommended text for the are, however, a number of differences, ranging from notation to omission, inclusion,and presentation of most topics. The heart of the notes is Chapter 6, which discusses thetheory of differentiation and all of its applications; the first five chapters essentially lay thenecessary mathematical foundation of analysis and linear far as prerequisites are concerned, I only assume that you are comfortable with allof the usual topics in single variable Calculus (limits, continuity, derivatives, optimization,integrals, sequences, series, Taylor polynomials, etc.)

4 In particular, Ido not assume any priorknowledge of linear algebra. Linear algebra naturally permeates the entire set of notes, but allof the necessary theory is introduced and explained. In general, I cover only the minimalamount of linear algebra needed, so it would be a bad idea to use these notes as a linearalgebra at the end of each section correspond to homework and exam problems I as-signed during the course. There are many topics that are standard in Multivariable calculuscourses (like the notion of projecting one vector onto another) that are introduced and stud-ied in the exercises, so keep this in mind. Challenging (optional) exercises are marked witha ( ). I ve also included some appendices that delve into more Advanced (optional) comments, corrections, or suggestions are welcome!

5 3 Chapter 1 Preliminariesteehee4 Chapter 2 Euclidean SpaceSingle variable Calculus is the study of functions of one variable. In slightly fancier lan-guage, single variable Calculus is the study of functionsf:R R. In order to studyfunctions of many variables which is the goal of Multivariable Calculus we first needto understand the underlying universe which hosts all of the forthcoming math. This uni-verse isEuclidean space, a generalization of the set of real numbersRtohigher dimensions(whatever that means!). This chapter introduces the basic algebra and geometry of Eu-clidean The elements of Euclidean spaceWe begin with the definition of Euclidean Euclidean space, read as R-n , as follows:Rn:={(x1,..,xn) :xj R}.Elements ofRnare calledvectors, and elements ofRare words,n-dimensional Euclidean space is the set of tuples ofnreal numbers, and bydefinition its elements are vectors.

6 You may have preconceived notions of a vector as beingan arrow, but you shouldn t really think of them this way at least, not initially. A vectoris just an element are a number of common notations used to represent vectors. For example,(x1,..,xn)and x1,..,xn and are all commonly used to denote an element ofRn. I will likely use all three notationsthroughout these notes. You may object:Joe, this is confusing! Why don t you just pick one andstick with it?To that, I have three answers:(i) I am too lazy to consistently stick with one notation.(ii) Sometimes, one notation is more convenient than another, depending on the contextof the problem.(iii) An important skill as a mathematician is to be able to recognize and adapt to unfamil-iar notation. There is almost never a universal notation for any mathematical object,and this will become apparent as you read more books, take more classes, and talk to5more mathematicians.

7 Learning to quickly recognize what one notation means in acertain context is extremely valuable!1In any case, here are some down to earth examples of (1,2) R2 0 1000 R3 500,0,0, e R4 One thing Iwillconsistently do in these notes is use boldface letters to indicate elementsofRn. For example, I might say: Letx Rnbe a vector. Letters that are not in boldfacewill usually denote I said that you shouldn t think of vectors as arrows, you actually can (andshould, sometimes) think of them that way. In particular, a vectorx= (x1,..,xn) Rnhas a geometric interpretation asthe arrow emanating from the origin(0,..,0)and terminatingat the coordinate(x1,..,xn).For example, inR2we could draw the vector(1,2)like this:(1,2)Sometimes it can be helpful to think of the vector(x1.)

8 ,xn)as an arrow with terminalpoint given by its coordinates, and other times it is better to think of it as simply its terminalpoint. However, I ll reiterate what I said above: a vector is simply an element ofRn. Anarrow is just a tool for The algebra of Euclidean spaceThe set of real numbersRcomes equipped with a number of algebraic operations likeaddition and multiplication, together with a host of rules like the distributive law thatgovern their interactions. Much of this algebraic structure extends naturally toRn, thoughsome of it is more subtle (like multiplication). Our first task is to establish some basicalgebraic definitions and rules in Euclidean ll begin by defining the notion ofvector additionandscalar (x1.

9 ,xn),y= (y1,..,yn) Rnbe vectors, and let Rbe ascalar.(i) Vector addition is defined as follows:x+y:= (x1+y1,..,xn+yn) day, you may even have to interact with a physicist. If this ever happens, you should be prepared toencounter some highly unusual (ii) Scalar multiplication is defined as follows: x:= ( x1,.., xn) words, to add two vectors (of the same dimension) you just add the correspondingcomponents, and to multiply a vector by a scalar you just multiply each component by thatscalar. Note that we have not yet defined how to multiply two vectors together; we ll talkabout this 123 = 246 10 1 + 223 = 322 If we interpret vectors as arrows emanating from the origin, vector addition and scalarmultiplication have nice geometric interpretations. In particular,- the vectorx+yis the main diagonal of the parallelogram generated byxandy, and- the vector xis the arrowx, stretched by a factor of.

10 See the figure below. One important consequence of the above two statements is that thevectorx yis the off diagonal of the parallelogram generated byxandy, travelling fromytox, shifted +yxy yx yx yThe following definition should be vectorsx,y Rnareparallelif there is a scalar Rsuch thatx= yor x= above discussion should make one thing clear: sometimes, it can be helpfulto visualize vectors as arrows emanating from points other than the origin. For example, itis natural to think ofx yas the off diagonal arrow, beginning at the terminal point I ll reiterate once more what I said above: a vector is simply an element ofRn. Drawingarrows is just a way to visualize such elements. In particular, you should really think ofallvectors as emanating from the conclude this section with a summary of all of the algebraic rules governing vec-tor addition and scalar multiplication.