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Lecture Notes for Advanced Calculus James S. Cook Liberty University Department of Mathematics Fall 2011

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introduction and motivations for these notes There are many excellent texts on portions of this subject. However, the particular path I choose this semester is not quite in line with any particular text. I required the text on Advanced Calculus by Edwards because it contains all the major theorems that traditionally are covered in an Advanced Calculus course. My focus differs significantly. If I had students who had already completed a semester in real analysis then we could delve into the more analytic aspects of the subject. However, real analysis is not a prerequisite so we take a different path. Generically the story is as follows: a linear approximation replaces a complicated object very well so long as we are close to the base-point for the approximation. The first level of understanding is what I would characterize as algebraic, beyond that is the analytic understanding. I would argue that we must first have a firm grasp of the algebraic before we can properly attack the analytic aspects of the subject. Edwards covers both the algebraic and the analytic. This makes his text hard to read in places because the full story is at some points technical. My goal is to focus on the algebraic. That said, I will try to at least point the reader to the section of Edward where the proof can be found. Linear algebra is not a prerequisite for this course. However, I will use linear algebra. Matrices, linear transformations and vector spaces are necessary ingredients for a proper discussion of advanced calculus. I believe an interested student can easily assimilate the needed tools as we go so I am not terribly worried if you have not had linear algebra previously. I will make a point to include some baby1 linear exercises to make sure everyone who is working at this course keeps up with the story that unfolds. Doing the homework is doing the course. I cannot overemphasize the importance of thinking through the homework. I would be happy if you left this course with a working knowledge of: ✓ set-theoretic mapping langauge, fibers and images and how to picture relationships diagramatically. ✓ continuity in view of the metric topology in n-space. ✓ the concept and application of the derivative and differential of a mapping. ✓ continuous differentiability ✓ inverse function theorem ✓ implicit function theorem ✓ tangent space and normal space via gradients 1

if you view this as an insult then you haven’t met the right babies yet. Baby exercises are cute.

3 ✓ extrema for multivariate functions, critical points and the Lagrange multiplier method ✓ multivariate Taylor series. ✓ quadratic forms ✓ critical point analysis for multivariate functions ✓ dual space and the dual basis. ✓ multilinear algebra. ✓ metric dualities and Hodge duality. ✓ the work and flux form mappings for ℝ3 . ✓ basic manifold theory ✓ vector fields as derivations. ✓ Lie series and how vector fields generate symmetries ✓ differential forms and the exterior derivative ✓ integration of forms ✓ generalized Stokes’s Theorem. ✓ surfaces ✓ fundmental forms and curvature for surfaces ✓ differential form formulation of classical differential geometry ✓ some algebra and calculus of supermathematics Before we begin, I should warn you that I assume quite a few things from the reader. These notes are intended for someone who has already grappled with the problem of constructing proofs. I assume you know the difference between ⇒ and ⇔. I assume the phrase ”iff” is known to you. I assume you are ready and willing to do a proof by induction, strong or weak. I assume you know what ℝ, ℂ, ℚ, ℕ and ℤ denote. I assume you know what a subset of a set is. I assume you know how to prove two sets are equal. I assume you are familar with basic set operations such as union and intersection (although we don’t use those much). More importantly, I assume you have started to appreciate that mathematics is more than just calculations. Calculations without context, without theory, are doomed to failure. At a minimum theory and proper mathematics allows you to communicate analytical concepts to other like-educated individuals. Some of the most seemingly basic objects in mathematics are insidiously complex. We’ve been taught they’re simple since our childhood, but as adults, mathematical adults, we find the actual

4 definitions of such objects as ℝ or ℂ are rather involved. I will not attempt to provide foundational arguments to build numbers from basic set theory. I believe it is possible, I think it’s well-thoughtout mathematics, but we take the existence of the real numbers as an axiom for these notes. We assume that ℝ exists and that the real numbers possess all their usual properties. In fact, I assume ℝ, ℂ, ℚ, ℕ and ℤ all exist complete with their standard properties. In short, I assume we have numbers to work with. We leave the rigorization of numbers to a different course. The format of these notes is similar to that of my calculus and linear algebra and advanced calculus notes from 2009-2011. However, I will make a number of definitions in the body of the text. Those sort of definitions are typically background-type definitions and I will make a point of putting them in bold so you can find them with ease. I have avoided use of Einstein’s implicit summation notation in the majority of these notes. This has introduced some clutter in calculations, but I hope the student finds the added detail helpful. Naturally if one goes on to study tensor calculations in physics then no such luxury is granted, you will have to grapple with the meaning of Einstein’s convention. I suspect that is a minority in this audience so I took that task off the to-do list for this course. The content of this course differs somewhat from my previous offering. The presentation of geometry and manifolds is almost entirely altered. Also, I have removed the chapter on Newtonian mechanics as well as the later chapter on variational calculus. Naturally, the interested student is invited to study those as indendent studies past this course. If interested please ask. I should mention that James Callahan’s Advanced Calculus: a geometric view has influenced my thinking in this reformulation of my notes. His discussion of Morse’s work was a useful addition to the critical point analysis. I was inspired by Flander’s text on differential form computation. It is my goal to implement some of his nicer calculations as an addition to my previous treatment of differential forms. In addition, I intend to encorporate material from Burns and Gidea’s Differential Geometry and Topology with a View to Dynamical Systems as well as Munkrese’ Analysis on Manifolds. These additions should greatly improve the depth of the manifold discussion. I intend to go significantly deeper this year so the student can perhaps begin to appreciate manifold theory. I plan to take the last few weeks of class to discuss supermathematics. This will serve as a sideways review for calculus on ℝ𝑛 . In addition, I hope the exercise of abstracting calculus to supernumbers gives you some ideas about the process of abstraction in general. Abstraction is a cornerstone of modern mathematics and it is an essential skill for a mathematician. We may also discuss some of the historical motivations and modern applications of supermath to supersymmetric field theory. NOTE To BE DELETED: -add pictures from 2009 notes.

5 -change equations to numbered equations.

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Contents 1 set-up 11 1.1 set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2 vectors and geometry for 𝑛-dimensional space . . . . . . . . . . . . . . . . . . . . . . 13 1.2.1 vector algebra for three dimensions . . . . . . . . . . . . . . . . . . . . . . . . 19 1.2.2 compact notations for vector arithmetic . . . . . . . . . . . . . . . . . . . . . 20 1.3 functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.4 elementary topology and limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2 linear algebra 37 2.1 vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.2 matrix calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.3 linear transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.3.1 a gallery of linear transformations . . . . . . . . . . . . . . . . . . . . . . . . 49 2.3.2 standard matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.3.3 coordinates and isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.4 normed vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3 differentiation 67 3.1 the differential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.2 partial derivatives and the existence of the differential . . . . . . . . . . . . . . . . . 73 3.2.1 directional derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.2.2 continuously differentiable, a cautionary tale . . . . . . . . . . . . . . . . . . 78 3.2.3 gallery of derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.3 additivity and homogeneity of the derivative . . . . . . . . . . . . . . . . . . . . . . . 86 3.4 chain rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.5 product rules? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.5.1 scalar-vector product rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.5.2 calculus of paths in ℝ3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.5.3 calculus of matrix-valued functions of a real variable . . . . . . . . . . . . . . 93 3.5.4 calculus of complex-valued functions of a real variable . . . . . . . . . . . . . 95 3.6 complex analysis in a nutshell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.6.1 harmonic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 7

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CONTENTS

4 inverse and implicit function theorems 103 4.1 inverse function theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.2 implicit function theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.3 implicit differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5 geometry of level sets 119 5.1 definition of level set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.2 tangents and normals to a level set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.3 method of Lagrange mulitpliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6 critical point analysis for several variables 133 6.1 multivariate power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.1.1 taylor’s polynomial for one-variable . . . . . . . . . . . . . . . . . . . . . . . . 133 6.1.2 taylor’s multinomial for two-variables . . . . . . . . . . . . . . . . . . . . . . 135 6.1.3 taylor’s multinomial for many-variables . . . . . . . . . . . . . . . . . . . . . 138 6.2 a brief introduction to the theory of quadratic forms . . . . . . . . . . . . . . . . . . 141 6.2.1 diagonalizing forms via eigenvectors . . . . . . . . . . . . . . . . . . . . . . . 144 6.3 second derivative test in many-variables . . . . . . . . . . . . . . . . . . . . . . . . . 151 6.3.1 morse theory and future reading . . . . . . . . . . . . . . . . . . . . . . . . . 154 7 multilinear algebra 155 7.1 dual space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 7.2 multilinearity and the tensor product . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 7.2.1 bilinear maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 7.2.2 trilinear maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 7.2.3 multilinear maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 7.3 wedge product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 7.3.1 wedge product of dual basis generates basis for Λ𝑉 . . . . . . . . . . . . . . . 167 7.3.2 the exterior algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 7.3.3 connecting vectors and forms in ℝ3 . . . . . . . . . . . . . . . . . . . . . . . . 175 7.4 bilinear forms and geometry; metric duality . . . . . . . . . . . . . . . . . . . . . . . 177 7.4.1 metric geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 7.4.2 metric duality for tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 7.4.3 inner products and induced norm . . . . . . . . . . . . . . . . . . . . . . . . . 182 7.5 hodge duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 7.5.1 hodge duality in euclidean space ℝ3 . . . . . . . . . . . . . . . . . . . . . . . 183 7.5.2 hodge duality in minkowski space ℝ4 . . . . . . . . . . . . . . . . . . . . . . . 185 7.6 coordinate change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 7.6.1 coordinate change for 𝑇20 (𝑉 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

CONTENTS

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8 manifold theory 193 8.1 manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 8.1.1 embedded manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 8.1.2 manifolds defined by charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 8.1.3 diffeomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 8.2 tangent space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 8.2.1 equivalence classes of curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 8.2.2 contravariant vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 8.2.3 derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 8.2.4 dictionary between formalisms . . . . . . . . . . . . . . . . . . . . . . . . . . 212 8.3 the differential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 8.4 cotangent space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 8.5 tensors at a point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 8.6 tensor fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 8.7 metric tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 8.7.1 classical metric notation in ℝ𝑚 . . . . . . . . . . . . . . . . . . . . . . . . . . 220 8.7.2 metric tensor on a smooth manifold . . . . . . . . . . . . . . . . . . . . . . . 221 8.8 on boundaries and submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 9 differential forms 229 9.1 algebra of differential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 9.2 exterior derivatives: the calculus of forms . . . . . . . . . . . . . . . . . . . . . . . . 231 9.2.1 coordinate independence of exterior derivative . . . . . . . . . . . . . . . . . . 232 9.2.2 exterior derivatives on ℝ3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 9.3 pullbacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 9.4 integration of differential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 9.4.1 integration of 𝑘-form on ℝ𝑘 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 9.4.2 orientations and submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 9.5 Generalized Stokes Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 9.6 poincare’s lemma and converse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 9.6.1 exact forms are closed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 9.6.2 potentials for closed forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 9.7 classical differential geometry in forms . . . . . . . . . . . . . . . . . . . . . . . . . . 257 9.8 E & M in differential form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 9.8.1 differential forms in Minkowski space . . . . . . . . . . . . . . . . . . . . . . . 258 9.8.2 exterior derivatives of charge forms, field tensors, and their duals . . . . . . 263 9.8.3 coderivatives and comparing to Griffith’s relativitic E & M . . . . . . . . . . 265 9.8.4 Maxwell’s equations are relativistically covariant . . . . . . . . . . . . . . . . 266 9.8.5 Electrostatics in Five dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 268 10 supermath

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CONTENTS

Chapter 1

set-up In this chapter we settle some basic terminology about sets and functions.

1.1

set theory

Let us denote sets by capital letters in as much as is possible. Often the lower-case letter of the same symbol will denote an element; 𝑎 ∈ 𝐴 is to mean that the object 𝑎 is in the set 𝐴. We can abbreviate 𝑎1 ∈ 𝐴 and 𝑎2 ∈ 𝐴 by simply writing 𝑎1 , 𝑎2 ∈ 𝐴, this is a standard notation. The union of two sets 𝐴 and 𝐵 is denoted 𝐴 ∪ 𝐵 = {𝑥∣𝑥 ∈ 𝐴 𝑜𝑟 𝑥 ∈ 𝐵}. The intersection of two sets is denoted 𝐴 ∩ 𝐵 = {𝑥∣𝑥 ∈ 𝐴 𝑎𝑛𝑑 𝑥 ∈ 𝐵}. If a set 𝑆 has no elements then we say 𝑆 is the empty set and denote this by writing 𝑆 = ∅. It sometimes convenient to use unions or intersections of several sets: ∪ 𝑈𝛼 = {𝑥 ∣ there exists 𝛼 ∈ Λ with 𝑥 ∈ 𝑈𝛼 } 𝛼∈Λ

∩

𝛼∈Λ

𝑈𝛼 = {𝑥 ∣ for all 𝛼 ∈ Λ we have 𝑥 ∈ 𝑈𝛼 }

we say Λ is the index set in the definitions above. If Λ is a finite set then the union/intersection is said to be a finite union/interection. If Λ is a countable set then the union/intersection is said to be a countable union/interection1 . Suppose 𝐴 and 𝐵 are both sets then we say 𝐴 is a subset of 𝐵 and write 𝐴 ⊆ 𝐵 iff 𝑎 ∈ 𝐴 implies 𝑎 ∈ 𝐵 for all 𝑎 ∈ 𝐴. If 𝐴 ⊆ 𝐵 then we also say 𝐵 is a superset of 𝐴. If 𝐴 ⊆ 𝐵 then we say 𝐴 ⊂ 𝐵 iff 𝐴 ∕= 𝐵 and 𝐴 ∕= ∅. Recall, for sets 𝐴, 𝐵 we define 𝐴 = 𝐵 iff 𝑎 ∈ 𝐴 implies 𝑎 ∈ 𝐵 for all 𝑎 ∈ 𝐴 and conversely 𝑏 ∈ 𝐵 implies 𝑏 ∈ 𝐴 for all 𝑏 ∈ 𝐵. This is equivalent to insisting 𝐴 = 𝐵 iff 𝐴 ⊆ 𝐵 and 𝐵 ⊆ 𝐴. The difference of two sets 𝐴 and 𝐵 is denoted 𝐴 − 𝐵 and is defined by 𝐴 − 𝐵 = {𝑎 ∈ 𝐴 ∣ such that 𝑎 ∈ / 𝐵}2 .

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recall the term countable simply means there exists a bijection to the natural numbers. The cardinality of such a set is said to be ℵ𝑜 2 other texts somtimes use 𝐴 − 𝐵 = 𝐴 ∖ 𝐵

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CHAPTER 1. SET-UP

We often make use of the following standard sets: natural numbers (positive integers); ℕ = {1, 2, 3, . . . }. natural numbers up to the number 𝑛; ℕ𝑛 = {1, 2, 3, . . . , 𝑛 − 1, 𝑛}. integers; ℤ = {. . . , −2, −1, 0, 1, 2, . . . }. Note, ℤ>0 = ℕ. non-negative integers; ℤ≥0 = {0, 1, 2, . . . } = ℕ ∪ {0}. negative integers; ℤ= 𝑎 < 1, 0, 0 > +𝑏 < 0, 1, 0 > +𝑐 < 0, 0, 1 >= 𝑎ˆ𝑖 + 𝑏ˆ𝑗 + 𝑐𝑘ˆ where we defined the ˆ𝑖 =< 1, 0, 0 >, ˆ𝑗 =< 0, 1, 0 >, ˆ𝑘 =< 0, 0, 1 >. You can easily verify that distinct Cartesian unit-vectors are orthogonal. Sometimes we need to produce a vector which is orthogonal to a given pair of vectors, it turns out the cross-product is one of two ways to do that in 𝑉 3 . We will see much later that this is special to three dimensions. Definition 1.2.16. If 𝐴 =< 𝐴1 , 𝐴2 , 𝐴3 > and 𝐵 =< 𝐵1 , 𝐵2 , 𝐵3 > are vectors in 𝑉 3 then the cross-product of 𝐴 and 𝐵 is a vector 𝐴 × 𝐵 which is defined by:  =< 𝐴2 𝐵3 − 𝐴3 𝐵2 , 𝐴3 𝐵1 − 𝐴1 𝐵3 , 𝐴1 𝐵2 − 𝐴2 𝐵1 > .  ×𝐵 𝐴  ×𝐵 can be shown to satisfy ∣∣ 𝐴  × 𝐵∣∣  = ∣∣𝐴∣∣  ∣∣𝐵∣∣  sin(𝜃) and the direction can The magnitude of 𝐴 be deduced by right-hand-rule. The right hand rule for the unit vectors yields: ˆ 𝑘ˆ × ˆ𝑖 = ˆ𝑗, ˆ𝑗 × 𝑘ˆ = ˆ𝑖 ˆ𝑖 × ˆ𝑗 = 𝑘, If I wish to discuss both the point and the vector to which it corresponds we may use the notatio...