DOCSLIB.ORG

Shanghai Lectures on Multivariable Analysis

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Shanghai Lectures on Multivariable Analysis Shanghai Lectures on Multivariable Analysis William G. Faris October 18, 2016 ii Contents 1 Differentiation 1 1.1 Fixed point iteration (single variable) . 2 1.2 The implicit function theorem (single variable) . 5 1.3 Linear algebra review (norms) . 7 1.4 Linear algebra review (eigenvalues) . 12 1.5 Differentiation (multivariable) . 16 1.6 Fixed point iteration (multivariable) . 26 1.7 The implicit function theorem (multivariable) . 28 1.8 Second order partial derivatives . 32 1.9 Problems . 34 2 Integration 43 2.1 The Riemann integral . 44 2.2 Jordan content . 48 2.3 Approximation of Riemann integrals . 48 2.4 Fubini's theorem . 50 2.5 Uniform convergence . 54 2.6 Dominated convergence . 55 2.7 Differentiating a parameterized integral . 58 2.8 Approximate delta functions . 60 2.9 Linear algebra (determinants) . 61 2.10 Change of variables . 64 2.11 Problems . 65 3 Differential Forms 71 3.1 Coordinates . 72 3.2 Scalar fields . 73 3.3 Vector fields . 74 3.4 Fluid velocity and the advective derivative . 78 3.5 Differential 1-forms . 79 3.6 Polar coordinates . 82 3.7 Integrating factors and canonical forms . 84 3.8 The second differential . 86 3.9 Regular surfaces . 88 iii iv CONTENTS 3.10 Lagrange multipliers . 91 3.11 Differential k-forms . 92 3.12 The exterior derivative . 94 3.13 The Poincar´elemma . 98 3.14 Substitution and pullback . 100 3.15 Pullback of a differential form . 102 3.16 Pushforward of a vector field . 104 3.17 Orientation . 106 3.18 Integration of top-dimension differential forms . 108 3.19 Integration of forms over singular surfaces . 109 3.20 Stokes' theorem for chains of singular surfaces . 110 3.21 Classical versions of Stokes' theorem . 113 3.22 Picturing Stokes' theorem . 115 3.23 Electric and magnetic fields . 119 4 The Metric Tensor 127 4.1 The interior product . 128 4.2 Volume . 129 4.3 The divergence theorem . 130 4.4 Conservation laws . 132 4.5 The metric . 135 4.6 Twisted forms . 138 4.7 The gradient and divergence and the Laplace operator . 139 4.8 Orthogonal coordinates and normalized bases . 141 4.9 Linear algebra (the Levi-Civita permutation symbol) . 144 4.10 Linear algebra (volume and area) . 145 4.11 Surface area . 148 5 Measure Zero 159 5.1 Outer content and outer measure . 160 5.2 The set of discontinuity of a function . 162 5.3 Lebesgue's theorem on Riemann integrability . 163 5.4 Almost everywhere . 165 5.5 Mapping sets of measure zero . 166 5.6 Sard's theorem . 167 5.7 Change of variables . 169 5.8 Fiber integration . 169 5.9 Probability . 171 5.10 The co-area formula . 172 5.11 Linear algebra (block matrices) . 175 Mathematical Notation 179 Preface The book This book originated in lectures given in Fall 2014 at NYU Shanghai for an advanced undergraduate course in multivariable analysis. There are chapters on Differentiation, Integration, Differential Forms, The Metric Tensor, together with an optional chapter on Measure Zero. The topics are standard, but the attempt is to present ideas that are often overlooked in this context. The fol- lowing chapter by chapter summary sketches the approach. The explanations in the summary are far from complete; they are only intended to highlight points that are explained in the body of the book. Differentiation This main themes of this chapter are standard. • It begins with fixed point iteration, the basis of the subsequent proofs. • The central object in this chapter is a smooth (that is, sufficiently differen- tiable) numerical function f defined on an open subset U of Rk with values in Rn. Here k is the domain dimension and n is the target dimension. The function is called \numerical" to emphasize that both inputs and outputs involve numbers. (See below for other kinds of functions.) Sometimes a numerical function is denoted by an expression like y 7! f(y). The choice of the variable name y is arbitrary. • When k < n a numerical function f from an open subset U of Rk to Rn can give a explicit (parametric) representation of a k-dimensional surface in Rn. • When k < n a numerical function g from an open subset W of Rn to Rn−k can given an implicit representation of a family of k-dimensional surfaces in W . • The implicit function theorem gives conditions for when an implicit rep- resentation gives rise to an explicit representation. v vi CONTENTS • When k = n the function f defines a transformation from an open subset U of Rn to an open set V in Rn. • The inverse function theorem gives conditions that ensure that the trans- formation has an inverse transformation. • In the passive interpretation the smooth transformation f : U ! V has a smooth inverse: the points in U and the points in V give alternative numerical descriptions of the same situation. • In the active interpretation the smooth transformation f : U ! U de- scribes a change of state: the state described by y in U is mapped into the new state described by f(y) in U. The transformation may be iterated. Integration This chapter is about the Riemann integral for functions of several variables. There are several interesting results. • The Fubini theorem for Riemann integrals deals with iterated integrals. • The dominated convergence theorem for Riemann integrals is a result about pointwise convergence. The setting is a sequence of Riemann in- tegrable functions defined on a fixed bounded set with a common bound on their values. The sequence of functions is assumed to converge (in some sense) to another Riemann integrable function. It is elementary to prove that if the functions converge uniformly, then the integrals converge. The dominated convergence theorem says that if the functions converge pointwise, then the integrals converge. The direct proof of the dominated convergence involves a somewhat complicated construction, but the result is spectacularly simple and useful. • There is a treatment of approximate delta functions. • The change of variables formula has an elegant proof via approximate delta functions and the dominated convergence theorem. Differential Forms This chapter and the next are the heart of the book. The central idea is ge- ometrical: differential forms are intrinsic expressions of change, and thus the basic mechanism of calculation with differential forms is equally simple for ev- ery possible choice of coordinate system. (Of course for modeling a particular system one coordinate system may be more convenient than another.) The fun- damental result is Stokes' theorem, which is the natural generalization of the fundamental theorem of calculus. Both the fundamental theorem in one dimen- sion and Stokes' theorem in higher dimensions make no reference to notions of length and area; they simply describe the cumulative effect of small changes. CONTENTS vii Because of this intrinsic nature, the expression of Stoke's theorem is the same in every coordinate system. • Example: Here is a simple example from physics that illustrates how nat- ural it is to have a free choice of coordinate system. Consider an idea gas with pressure P , volume V , and temperature T . The number of gas par- ticles N is assumed constant. Each of the quantities P; V; T is a function of the state of the gas; these functions are related by the ideal gas law PV = NkT: (The constant k transforms temperature units to energy units.) The dif- ferential form of this relation is P dV + V dP = Nk dT: This equation is a precise description of the how these variables change for a small change in the state of the gas. A typical use of the fundamental theorem of calculus is the calculation of work done by the system during a change of state where the temperature is constant. This is obtained by integrating −P dV along states of constant temperature T = T0. Suppose that in this process the volume changes from V0 to V1 and the pressure changes from P0 to P1. Then −P dV = −NkT0 dV=V = −NkT0 d log(V ) has integral NkT0 log(V0=V1). But nothing depends on using volume as an independent variable. On the curve where T = T0 there is a relation P dV + V dP = 0. So on this curve the form is also equal to V dP = NkT0 dP=P = NkT0 log(P ) with integral NkT0 log(P1=P0). Since P0V0 = P1V1 = NkT0, this is the same result. • While many applications of differential forms have nothing to do with length or area, others make essential use of these ideas. For instance, the length of a curve in the plane is obtained by integrating p(dx=dt)2 + (dy=dt)2 dt = p(dx=du)2 + (dy=du)2 du: The motion along the curve may be described either by the t coordinate or the u coordinate, but the length of the curve between two points is a num- ber that does not depend on this choice. In practice such integrals involv- ing square roots of sums of squares can be awkward. Sometimes it helps to use a different coordinate system in the plane. For instance, in polar coordinates the same form has the expression p(dr=dt)2 + r2(dθ=dt)2 dt. The theory of differential forms gives a systematic way of dealing with all such coordinate changes. viii CONTENTS • The chapter introduces a fundamental organizing principle of mathemat- ical modeling. It applies in geometry and in most applications of mathe- matics, and it deserves a serious discussion. In the approach of this book it is based on the elementary concept of an n-dimensional manifold patch. This term is used here for a differentiable manifold M modeled on some open subset of Rn.
Load more

Recommended publications
  • Lecture 11 Line Integrals of Vector-Valued Functions (Cont'd)
    Lecture 11 Line integrals of vector-valued functions (cont’d) In the previous lecture, we considered the following physical situation: A force, F(x), which is not necessarily constant in space, is acting on a mass m, as the mass moves along a curve C from point P to point Q as shown in the diagram below. z Q P m F(x(t)) x(t) C y x The goal is to compute the total amount of work W done by the force. Clearly the constant force/straight line displacement formula, W = F · d , (1) where F is force and d is the displacement vector, does not apply here. But the fundamental idea, in the “Spirit of Calculus,” is to break up the motion into tiny pieces over which we can use Eq. (1 as an approximation over small pieces of the curve. We then “sum up,” i.e., integrate, over all contributions to obtain W . The total work is the line integral of the vector field F over the curve C: W = F · dx . (2) ZC Here, we summarize the main steps involved in the computation of this line integral: 1. Step 1: Parametrize the curve C We assume that the curve C can be parametrized, i.e., x(t)= g(t) = (x(t),y(t), z(t)), t ∈ [a, b], (3) so that g(a) is point P and g(b) is point Q. From this parametrization we can compute the velocity vector, ′ ′ ′ ′ v(t)= g (t) = (x (t),y (t), z (t)) . (4) 1 2. Step 2: Compute field vector F(g(t)) over curve C F(g(t)) = F(x(t),y(t), z(t)) (5) = (F1(x(t),y(t), z(t), F2(x(t),y(t), z(t), F3(x(t),y(t), z(t))) , t ∈ [a, b] .
    [Show full text]
  • Notes on Partial Differential Equations John K. Hunter
    Notes on Partial Differential Equations John K. Hunter Department of Mathematics, University of California at Davis Contents Chapter 1. Preliminaries 1 1.1. Euclidean space 1 1.2. Spaces of continuous functions 1 1.3. H¨olderspaces 2 1.4. Lp spaces 3 1.5. Compactness 6 1.6. Averages 7 1.7. Convolutions 7 1.8. Derivatives and multi-index notation 8 1.9. Mollifiers 10 1.10. Boundaries of open sets 12 1.11. Change of variables 16 1.12. Divergence theorem 16 Chapter 2. Laplace's equation 19 2.1. Mean value theorem 20 2.2. Derivative estimates and analyticity 23 2.3. Maximum principle 26 2.4. Harnack's inequality 31 2.5. Green's identities 32 2.6. Fundamental solution 33 2.7. The Newtonian potential 34 2.8. Singular integral operators 43 Chapter 3. Sobolev spaces 47 3.1. Weak derivatives 47 3.2. Examples 47 3.3. Distributions 50 3.4. Properties of weak derivatives 53 3.5. Sobolev spaces 56 3.6. Approximation of Sobolev functions 57 3.7. Sobolev embedding: p < n 57 3.8. Sobolev embedding: p > n 66 3.9. Boundary values of Sobolev functions 69 3.10. Compactness results 71 3.11. Sobolev functions on Ω ⊂ Rn 73 3.A. Lipschitz functions 75 3.B. Absolutely continuous functions 76 3.C. Functions of bounded variation 78 3.D. Borel measures on R 80 v vi CONTENTS 3.E. Radon measures on R 82 3.F. Lebesgue-Stieltjes measures 83 3.G. Integration 84 3.H. Summary 86 Chapter 4.
    [Show full text]
  • MULTIVARIABLE CALCULUS (MATH 212 ) COMMON TOPIC LIST (Approved by the Occidental College Math Department on August 28, 2009)
    MULTIVARIABLE CALCULUS (MATH 212 ) COMMON TOPIC LIST (Approved by the Occidental College Math Department on August 28, 2009) Required Topics • Multivariable functions • 3-D space o Distance o Equations of Spheres o Graphing Planes Spheres Miscellaneous function graphs Sections and level curves (contour diagrams) Level surfaces • Planes o Equations of planes, in different forms o Equation of plane from three points o Linear approximation and tangent planes • Partial Derivatives o Compute using the definition of partial derivative o Compute using differentiation rules o Interpret o Approximate, given contour diagram, table of values, or sketch of sections o Signs from real-world description o Compute a gradient vector • Vectors o Arithmetic on vectors, pictorially and by components o Dot product compute using components v v v v v ⋅ w = v w cosθ v v v v u ⋅ v = 0⇔ u ⊥ v (for nonzero vectors) o Cross product Direction and length compute using components and determinant • Directional derivatives o estimate from contour diagram o compute using the lim definition t→0 v v o v compute using fu = ∇f ⋅ u ( u a unit vector) • Gradient v o v geometry (3 facts, and understand how they follow from fu = ∇f ⋅ u ) Gradient points in direction of fastest increase Length of gradient is the directional derivative in that direction Gradient is perpendicular to level set o given contour diagram, draw a gradient vector • Chain rules • Higher-order partials o compute o mixed partials are equal (under certain conditions) • Optimization o Locate and
    [Show full text]
  • Engineering Analysis 2 : Multivariate Functions
    Multivariate functions Partial Differentiation Higher Order Partial Derivatives Total Differentiation Line Integrals Surface Integrals Engineering Analysis 2 : Multivariate functions P. Rees, O. Kryvchenkova and P.D. Ledger, [email protected] College of Engineering, Swansea University, UK PDL (CoE) SS 2017 1/ 67 Multivariate functions Partial Differentiation Higher Order Partial Derivatives Total Differentiation Line Integrals Surface Integrals Outline 1 Multivariate functions 2 Partial Differentiation 3 Higher Order Partial Derivatives 4 Total Differentiation 5 Line Integrals 6 Surface Integrals PDL (CoE) SS 2017 2/ 67 Multivariate functions Partial Differentiation Higher Order Partial Derivatives Total Differentiation Line Integrals Surface Integrals Introduction Recall from EG189: analysis of a function of single variable: Differentiation d f (x) d x Expresses the rate of change of a function f with respect to x. Integration Z f (x) d x Reverses the operation of differentiation and can used to work out areas under curves, and as we have just seen to solve ODEs. We’ve seen in vectors we often need to deal with physical fields, that depend on more than one variable. We may be interested in rates of change to each coordinate (e.g. x; y; z), time t, but sometimes also other physical fields as well. PDL (CoE) SS 2017 3/ 67 Multivariate functions Partial Differentiation Higher Order Partial Derivatives Total Differentiation Line Integrals Surface Integrals Introduction (Continue ...) Example 1: the area A of a rectangular plate of width x and breadth y can be calculated A = xy The variables x and y are independent of each other. In this case, the dependent variable A is a function of the two independent variables x and y as A = f (x; y) or A ≡ A(x; y) PDL (CoE) SS 2017 4/ 67 Multivariate functions Partial Differentiation Higher Order Partial Derivatives Total Differentiation Line Integrals Surface Integrals Introduction (Continue ...) Example 2: the volume of a rectangular plate is given by V = xyz where the thickness of the plate is in z-direction.
    [Show full text]
  • 33 .1 Implicit Differentiation 33.1.1Implicit Function
    Module 11 : Partial derivatives, Chain rules, Implicit differentiation, Gradient, Directional derivatives Lecture 33 : Implicit differentiation [Section 33.1] Objectives In this section you will learn the following : The concept of implicit differentiation of functions. 33 .1 Implicit differentiation \ As we had observed in section , many a times a function of an independent variable is not given explicitly, but implicitly by a relation. In section we had also mentioned about the implicit function theorem. We state it precisely now, without proof. 33.1.1Implicit Function Theorem (IFT): Let and be such that the following holds: (i) Both the partial derivatives and exist and are continuous in . (ii) and . Then there exists some and a function such that is differentiable, its derivative is continuous with and 33.1.2Remark: We have a corresponding version of the IFT for solving in terms of . Here, the hypothesis would be . 33.1.3Example: Let we want to know, when does the implicit expression defines explicitly as a function of . We note that and are both continuous. Since for the points and the implicit function theorem is not applicable. For , and , the equation defines the explicit function and for , the equation defines the explicit function Figure 1. y is a function of x. A result similar to that of theorem holds for function of three variables, as stated next. Theorem : 33.1.4 Let and be such that (i) exist and are continuous at . (ii) and . Then the equation determines a unique function in the neighborhood of such that for , and , Practice Exercises : Show that the following functions satisfy conditions of the implicit function theorem in the neighborhood of (1) the indicated point.
    [Show full text]
  • Partial Derivatives
    Partial Derivatives Partial Derivatives Just as derivatives can be used to explore the properties of functions of 1 vari- able, so also derivatives can be used to explore functions of 2 variables. In this section, we begin that exploration by introducing the concept of a partial derivative of a function of 2 variables. In particular, we de…ne the partial derivative of f (x; y) with respect to x to be f (x + h; y) f (x; y) fx (x; y) = lim h 0 h ! when the limit exists. That is, we compute the derivative of f (x; y) as if x is the variable and all other variables are held constant. To facilitate the computation of partial derivatives, we de…ne the operator @ = “The partial derivative with respect to x” @x Alternatively, other notations for the partial derivative of f with respect to x are @f @ f (x; y) = = f (x; y) = @ f (x; y) x @x @x x 2 2 EXAMPLE 1 Evaluate fx when f (x; y) = x y + y : Solution: To do so, we write @ @ @ f (x; y) = x2y + y2 = x2y + y2 x @x @x @x and then we evaluate the derivative as if y is a constant. In partic- ular, @ 2 @ 2 fx (x; y) = y x + y = y 2x + 0 @x @x That is, y factors to the front since it is considered constant with respect to x: Likewise, y2 is considered constant with respect to x; so that its derivative with respect to x is 0. Thus, fx (x; y) = 2xy: Likewise, the partial derivative of f (x; y) with respect to y is de…ned f (x; y + h) f (x; y) fy (x; y) = lim h 0 h ! 1 when the limit exists.
    [Show full text]
  • Partial Differential Equations a Partial Difierential Equation Is a Relationship Among Partial Derivatives of a Function
    In contrast, ordinary differential equations have only one independent variable. Often associate variables to coordinates (Cartesian, polar, etc.) of a physical system, e.g. u(x; y; z; t) or w(r; θ) The domain of the functions involved (usually denoted Ω) is an essential detail. Often other conditions placed on domain boundary, denoted @Ω. Partial differential equations A partial differential equation is a relationship among partial derivatives of a function (or functions) of more than one variable. Often associate variables to coordinates (Cartesian, polar, etc.) of a physical system, e.g. u(x; y; z; t) or w(r; θ) The domain of the functions involved (usually denoted Ω) is an essential detail. Often other conditions placed on domain boundary, denoted @Ω. Partial differential equations A partial differential equation is a relationship among partial derivatives of a function (or functions) of more than one variable. In contrast, ordinary differential equations have only one independent variable. The domain of the functions involved (usually denoted Ω) is an essential detail. Often other conditions placed on domain boundary, denoted @Ω. Partial differential equations A partial differential equation is a relationship among partial derivatives of a function (or functions) of more than one variable. In contrast, ordinary differential equations have only one independent variable. Often associate variables to coordinates (Cartesian, polar, etc.) of a physical system, e.g. u(x; y; z; t) or w(r; θ) Often other conditions placed on domain boundary, denoted @Ω. Partial differential equations A partial differential equation is a relationship among partial derivatives of a function (or functions) of more than one variable.
    [Show full text]
  • Notes for Math 136: Review of Calculus
    Notes for Math 136: Review of Calculus Gyu Eun Lee These notes were written for my personal use for teaching purposes for the course Math 136 running in the Spring of 2016 at UCLA. They are also intended for use as a general calculus reference for this course. In these notes we will briefly review the results of the calculus of several variables most frequently used in partial differential equations. The selection of topics is inspired by the list of topics given by Strauss at the end of section 1.1, but we will not cover everything. Certain topics are covered in the Appendix to the textbook, and we will not deal with them here. The results of single-variable calculus are assumed to be familiar. Practicality, not rigor, is the aim. This document will be updated throughout the quarter as we encounter material involving additional topics. We will focus on functions of two real variables, u = u(x;t). Most definitions and results listed will have generalizations to higher numbers of variables, and we hope the generalizations will be reasonably straightforward to the reader if they become necessary. Occasionally (notably for the chain rule in its many forms) it will be convenient to work with functions of an arbitrary number of variables. We will be rather loose about the issues of differentiability and integrability: unless otherwise stated, we will assume all derivatives and integrals exist, and if we require it we will assume derivatives are continuous up to the required order. (This is also generally the assumption in Strauss.) 1 Differential calculus of several variables 1.1 Partial derivatives Given a scalar function u = u(x;t) of two variables, we may hold one of the variables constant and regard it as a function of one variable only: vx(t) = u(x;t) = wt(x): Then the partial derivatives of u with respect of x and with respect to t are defined as the familiar derivatives from single variable calculus: ¶u dv ¶u dw = x ; = t : ¶t dt ¶x dx c 2016, Gyu Eun Lee.
    [Show full text]
  • Multivariable and Vector Calculus
    Multivariable and Vector Calculus Lecture Notes for MATH 0200 (Spring 2015) Frederick Tsz-Ho Fong Department of Mathematics Brown University Contents 1 Three-Dimensional Space ....................................5 1.1 Rectangular Coordinates in R3 5 1.2 Dot Product7 1.3 Cross Product9 1.4 Lines and Planes 11 1.5 Parametric Curves 13 2 Partial Differentiations ....................................... 19 2.1 Functions of Several Variables 19 2.2 Partial Derivatives 22 2.3 Chain Rule 26 2.4 Directional Derivatives 30 2.5 Tangent Planes 34 2.6 Local Extrema 36 2.7 Lagrange’s Multiplier 41 2.8 Optimizations 46 3 Multiple Integrations ........................................ 49 3.1 Double Integrals in Rectangular Coordinates 49 3.2 Fubini’s Theorem for General Regions 53 3.3 Double Integrals in Polar Coordinates 57 3.4 Triple Integrals in Rectangular Coordinates 62 3.5 Triple Integrals in Cylindrical Coordinates 67 3.6 Triple Integrals in Spherical Coordinates 70 4 Vector Calculus ............................................ 75 4.1 Vector Fields on R2 and R3 75 4.2 Line Integrals of Vector Fields 83 4.3 Conservative Vector Fields 88 4.4 Green’s Theorem 98 4.5 Parametric Surfaces 105 4.6 Stokes’ Theorem 120 4.7 Divergence Theorem 127 5 Topics in Physics and Engineering .......................... 133 5.1 Coulomb’s Law 133 5.2 Introduction to Maxwell’s Equations 137 5.3 Heat Diffusion 141 5.4 Dirac Delta Functions 144 1 — Three-Dimensional Space 1.1 Rectangular Coordinates in R3 Throughout the course, we will use an ordered triple (x, y, z) to represent a point in the three dimensional space.
    [Show full text]
  • Integrated Calculus/Pre-Calculus
    Furman University Department of Mathematics Greenville, SC, 29613 Integrated Calculus/Pre-Calculus Abstract Mark R. Woodard Contents JJ II J I Home Page Go Back Close Quit Abstract This work presents the traditional material of calculus I with some of the material from a traditional precalculus course interwoven throughout the discussion. Pre- calculus topics are discussed at or soon before the time they are needed, in order to facilitate the learning of the calculus material. Miniature animated demonstra- tions and interactive quizzes will be available to help the reader deepen his or her understanding of the material under discussion. Integrated This project is funded in part by the Mellon Foundation through the Mellon Furman- Calculus/Pre-Calculus Wofford Program. Many of the illustrations were designed by Furman undergraduate Mark R. Woodard student Brian Wagner using the PSTricks package. Thanks go to my wife Suzan, and my two daughters, Hannah and Darby for patience while this project was being produced. Title Page Contents JJ II J I Go Back Close Quit Page 2 of 191 Contents 1 Functions and their Properties 11 1.1 Functions & Cartesian Coordinates .................. 13 1.1.1 Functions and Functional Notation ............ 13 1.2 Cartesian Coordinates and the Graphical Representa- tion of Functions .................................. 20 1.3 Circles, Distances, Completing the Square ............ 22 1.3.1 Distance Formula and Circles ................. 22 Integrated 1.3.2 Completing the Square ...................... 23 Calculus/Pre-Calculus 1.4 Lines ............................................ 24 Mark R. Woodard 1.4.1 General Equation and Slope-Intercept Form .... 24 1.4.2 More On Slope ............................. 24 1.4.3 Parallel and Perpendicular Lines .............
    [Show full text]
  • Calculus Terminology
    AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential
    [Show full text]
  • Vector Calculus and Differential Forms with Applications To
    Vector Calculus and Differential Forms with Applications to Electromagnetism Sean Roberson May 7, 2015 PREFACE This paper is written as a final project for a course in vector analysis, taught at Texas A&M University - San Antonio in the spring of 2015 as an independent study course. Students in mathematics, physics, engineering, and the sciences usually go through a sequence of three calculus courses before go- ing on to differential equations, real analysis, and linear algebra. In the third course, traditionally reserved for multivariable calculus, stu- dents usually learn how to differentiate functions of several variable and integrate over general domains in space. Very rarely, as was my case, will professors have time to cover the important integral theo- rems using vector functions: Green’s Theorem, Stokes’ Theorem, etc. In some universities, such as UCSD and Cornell, honors students are able to take an accelerated calculus sequence using the text Vector Cal- culus, Linear Algebra, and Differential Forms by John Hamal Hubbard and Barbara Burke Hubbard. Here, students learn multivariable cal- culus using linear algebra and real analysis, and then they generalize familiar integral theorems using the language of differential forms. This paper was written over the course of one semester, where the majority of the book was covered. Some details, such as orientation of manifolds, topology, and the foundation of the integral were skipped to save length. The paper should still be readable by a student with at least three semesters of calculus, one course in linear algebra, and one course in real analysis - all at the undergraduate level.
    [Show full text]