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MULTIVARIABLE ANALYSIS What Follows Are Lecture Notes from An MULTIVARIABLE ANALYSIS DANIEL S. FREED What follows are lecture notes from an advanced undergraduate course given at the University of Texas at Austin in Spring, 2019. The notes are rough in many places, so use at your own risk! Contents Lecture 1: Affine geometry 4 Basic definitions4 Affine analogs of vector space concepts6 Ceva's theorem8 Lecture 2: Parallelism, coordinates, and symmetry 10 Parallelism and another classical theorem 10 Bases in vector spaces 11 Linear symmetry groups 12 Affine coordinates 13 The tangent space to affine space 15 Lecture 3. Normed linear spaces 16 Basic definitions 16 Norms on Rn 17 Lecture 4. More on normed linear spaces 20 Banach spaces 20 Examples of Banach spaces 20 Equivalent norms 22 Lecture 5. Continuous linear maps; differentiability 24 Continuous linear maps 24 Shapes and functions 26 Setting for calculus 27 Continuity and differentiability 29 Lecture 6: Computation of the differential 31 Differentiability and continuity 31 Functions of one variable 31 Computation of the differential 32 The operator d and explicit computation 34 Lecture 7: Further properties of the differential 36 Chain rule 36 Mean value inequality 39 Date: February 2, 2021. 1 2 D. S. FREED Lecture 8: Differentials and local extrema; inner products 41 Continuous differentiability 41 Local extrema and critical points 43 Real inner product spaces 45 Lecture 9: Inner product spaces; gradients 47 More on inner product spaces 47 Gradient 49 Lecture 10: Differential of the length function; contraction mappings 51 The first variation formula 51 Fixed points and contractions 56 Lecture 11: Inverse function theorem 58 Invertibility is an open condition 58 Ck functions 59 Inverse function theorem 60 Lecture 12: Examples and computations 64 Characterization of affine maps 64 Examples of the inverse function theorem 64 Sample computations 66 Lecture 13: Implicit function theorem; second differential 68 Motivation and examples 68 Implicit function theorem 71 Introduction to the second differential 72 Lecture 14: The second differential; symmetric bilinear forms 74 Main theorems about the second differential 74 Symmetric bilinear forms 77 Lecture 15: Quadratic approximation and the second variation formula 80 Symmetric bilinear forms and inner products 80 The second derivative test and quadratic approximation 81 Second variation formula 83 Lecture 16: Introduction to curvature 85 Introduction 85 Curvature of a curve 86 Curvature of a surface in Euclidean 3-space 92 Lecture 17: Fundamental theorem of ordinary differential equations 96 Setup and Motivation 96 The Riemann integral 97 An application of the contraction mapping fixed point theorem 99 Lecture 18: More on ODE; introduction to differential forms 102 Uniqueness and global solutions 102 Additional topics 104 Motivating differential forms, part one 105 Lecture 19: Integration along curves and surfaces; universal properties 108 Motivating differential forms, part two 108 Bonus material: integration of 1-forms 111 MULTIVARIABLE ANALYSIS 3 Characterization by a universal property 115 Lecture 20: Tensor products, tensor algebras, and exterior algebras 119 Tensor products of vector spaces 119 Tensor algebra 122 Exterior algebra 124 Lecture 21: More on the exterior algebra; determinants 126 More about the exterior algebra: Z-grading and commutativity 126 Direct sums and tensor products 128 Finite dimensional exterior algebras and determinants 131 Lecture 22: Orientation and signed volume 134 Orientations 134 Duality and exterior algebras 136 Signed volume 137 Lecture 23: The Cartan exterior differential 139 Introduction 139 Exterior d 139 Lecture 24: Pullback of differential forms; forms on bases 144 Pullbacks 144 Bases of vector spaces and differential forms 146 Bases of affine spaces and differential forms 148 Frames on Euclidean space 149 Lecture 25: Curvature of curves and surfaces 153 Curvature of plane curves 153 Curvature of surfaces 155 Lectures 26{28: Integration 159 Integration of differential forms 161 4 D. S. FREED Lecture 1: Affine geometry Basic definitions Definition 1.1. Let V be a vector space. An affine space A over V is a set A with a simply transitive action of V . Elements of A are called points; elements of V are called vectors. The result of the action of a vector ξ P V on a point p P A is written p ` ξ P A. We call V the tangent space to A, and will explain the nomenclature in the next lecture; see( 2.33). a Figure 1. An affine space A over a vector space V Remark 1.2 (Data and conditions). A vector space pV; 0; `; ˚q over a field F consists of four pieces of data: a set V , a distinguished element 0 P V , a function `: V ˆ V Ñ V , and a function ˚: F ˆ V Ñ V . These data satisfy several axioms or conditions, including the fact that 0 is an identity for `; the existence of inverses; and commutativity, associativity, and distributivity axioms. An affine space pA; `q over V provides two additional pieces of data: a set A and a function (1.3) `: A ˆ V Ñ A: (This overloading of the symbol ``' should not cause trouble.) There are additional axioms regard- ing the new `, here encoded in the phrase `simply transitive action'. As usual, symbols like `V ' and `A' invoke all constituents of the relevant structure. (1.4) Simple transitivity. A vector space pV; 0; `; ˚q determines an abelian group pV; 0; `q. It is this abelian group which acts simply transitively on A. Simple transitivity is the statement that the map A ˆ V ÝÑ A ˆ A (1.5) p ; ξ ÞÝÑ p ; p ` ξ is a bijection. Hence given p0; p1 P A there exists a unique ξ P V such that p1 “ p0 ` ξ. We write ξ “ p1 ´ p0. MULTIVARIABLE ANALYSIS 5 Remark 1.6. In this course we will almost exclusively consider F “ R. At the moment there is no topology on either V or A, and indeed much of the affine geometry we discuss in the first few lectures is valid over any field. Soon we will introduce an additional structure on V |a norm| which will induce a topology and allow us to discuss limits, completeness, compactness, and other topological properties which we use to develop analysis. Remark 1.7. Affine space is the arena for flat geometry. The geometry of flat space was studied by Euclid and his contemporaries, but in Euclidean geometry there is more structure: distance and angle. The second half of the word `geometry' invokes `measurement', albeit measurement of the earth (`geo'), which presumably was the (flawed) model for this part of ancient mathematics. We will discuss geometric structures in affine geometry, such as a Euclidean structure, but for now there is no measurement in affine geometry. (However, see( 1.15) below.) Therefore ‘affine geometry' is an oxymoron: the earth is not flat and measurement is not possible in bare affine space. The distinction between points and vectors is more obvious in curved spaces, but even in flat geometry they play very different roles. For example, we might model time by an affine space A over a one-dimensional vector space V . Points of A represent instants of time, whereas elements in the group V represent intervals of time. Notice that it makes sense to add intervals of time, say 3 hours ` 4 hours “ 7 hours, whereas 3:00 AM ` 4:00 PM does not make good sense. Then again, that the difference 4:00 PM ´ 3:00 PM (the same day!) is the time interval 1 hour is part of our intuition about time. Similarly, in a flat model of the Earth we do not try to make sense of the sum of Chicago and New York, but their difference as a displacement vector does make sense. (1.8) Points vs. vectors. A vector space V has a canonical (trivial) affine space over it defined by setting A “ V and letting (1.3) be vector addition. One can loosely describe this as \forgetting the zero vector". (1.9) Vector spaces as affine spaces. Definition 1.10. Let A be affine over a vector space V and B affine over a vector space W . Then a function f : A Ñ B is an affine map if there exists a linear map T : V Ñ W such that (1.11) fpp ` ξq “ fppq ` T ξ for all p P A, ξ P V . The linear map T is called the differential of f, and we write T “ df. It is easy to verify that T is unique, if it exists. We will soon study non-affine maps A Ñ B, and such a map is differentiable if for each p there exists a linear Tp|its differential at p|such that (1.11) holds up to a controlled error term. An affine map has a constant differential. Affine geometry is the study of affine spaces and affine maps between them. The function τξ0 A Ñ A which results from (1.3) by freezing ξ0 P V is called translation by ξ0. It is an affine automorphism of A. On the other hand, if we freeze p0 P A then we obtain an isomorphism of affine spaces θp0 : V ÝÑ A (1.13) ξ ÞÝÑ p ` ξ 6 D. S. FREED ´1 Given a second point p1 P A the transition function θp1 ˝ θp0 : V Ñ V is translation by p0 ´ p1. In other words, we can identify an affine space with its tangent space up to a translation. (1.12) Structural affine isomorphisms. If A is affine over V , then we denote the group of affine automorphisms A Ñ A as AutpAq, and the group of linear automorphisms of V as AutpV q. There is a short exact sequence of groups τ d (1.14) 1 ÝÑ V ÝÝÑ AutpAq ÝÝÑ AutpV q ÝÑ 1 Thus, d ˝ τ is the constant map onto idV and moreover ker d “ τpV q; in other words, an affine map f satisfies df “ idV if and only if f is a translation.
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