DOCSLIB.ORG

MAT 169: Calculus III with Analytic Geometry

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
MAT 169: Calculus III with Analytic Geometry MAT 169: Calculus III with Analytic Geometry James V. Lambers December 7, 2012 2 Contents 1 Sequences and Series 7 1.1 Introduction . 7 1.1.1 Sequences and Series . 7 1.1.2 Vectors and the Geometry of Space . 8 1.1.3 Parametric Equations and Polar Coordinates . 9 1.2 Review of Calculus . 10 1.2.1 Limits . 10 1.2.2 Continuity . 11 1.2.3 Derivatives . 12 1.2.4 Riemann Sums and the Definite Integral . 13 1.2.5 Extreme Values . 14 1.2.6 The Mean Value Theorem . 16 1.2.7 The Mean Value Theorem for Integrals . 17 1.3 Taylor's Theorem . 18 1.4 Sequences . 21 1.4.1 What is a Sequence? . 21 1.4.2 Why Do We Need Sequences? . 23 1.4.3 Recognizing Sequences . 23 1.4.4 Limits of Sequences . 24 1.4.5 Relation to Limits of Functions . 26 1.4.6 Testing Convergence of Sequences . 27 1.4.7 Alternating Sequences . 28 1.4.8 Growth Rates of Functions . 29 1.4.9 Geometric Sequences . 30 1.4.10 Recursively Defined Sequences . 30 1.4.11 Bounded and Monotonic Sequences . 31 1.4.12 Summary . 35 1.5 Series . 37 1.5.1 What is a Series? . 37 3 4 CONTENTS 1.5.2 Why Do We Need Series? . 39 1.5.3 Geometric Series . 40 1.5.4 Telescoping Series . 43 1.5.5 Harmonic Series . 44 1.5.6 Basic Convergence Tests . 44 1.5.7 Summary . 45 1.6 The Integral and Comparison Tests . 46 1.6.1 The Integral Test . 46 1.6.2 The Comparison Test . 51 1.7 Other Convergence Tests . 54 1.7.1 The Alternating Series Test . 55 1.7.2 Estimating Error in Alternating Series . 56 1.7.3 Absolute Convergence . 58 1.7.4 The Ratio Test . 59 1.7.5 The Root Test . 60 1.7.6 Summary . 61 1.8 Power Series . 62 1.8.1 What is a Power Series? . 62 1.8.2 Convergence of Power Series . 63 1.8.3 The Radius of Convergence . 64 1.8.4 Representing Functions as Power Series . 64 1.8.5 Differentiation and Integration of Power Series . 66 1.8.6 Summary . 68 1.9 Taylor and Maclaurin Series . 70 1.9.1 Summary . 81 1.10 Review . 83 2 Vectors and the Geometry of Space 87 2.1 Three-Dimensional Coordinate Systems . 87 2.1.1 Points in Three-Dimensional Space . 87 2.1.2 Planes in Three-Dimensional Space . 88 2.1.3 Plotting Points in xyz-space . 89 2.1.4 The Distance Formula . 89 2.1.5 Equations of Surfaces . 90 2.1.6 Summary . 92 2.2 Vectors . 93 2.2.1 Combining Vectors . 94 2.2.2 Components . 95 2.2.3 Summary . 102 2.3 The Dot Product . 103 CONTENTS 5 2.3.1 Properties . 105 2.3.2 Orthogonality . 106 2.3.3 Projections . 106 2.3.4 Summary . 109 2.4 The Cross Product . 110 2.4.1 Parallel Vectors . 113 2.4.2 Properties . 113 2.4.3 Areas . 113 2.4.4 Volumes . 115 2.4.5 Summary . 115 2.5 Equations of Lines and Planes . 116 2.5.1 Equations of Lines . 116 2.5.2 Systems of Linear Equations . 120 2.5.3 Equations of Planes . 126 2.5.4 Intersecting Planes . 128 2.5.5 Distance from a Point to a Plane . 129 2.5.6 Summary . 131 2.6 Review . 132 3 Parametric Curves and Polar Coordinates 143 3.1 Parametric Curves . 143 3.1.1 Summary . 147 3.2 Calculus with Parametric Curves . 148 3.2.1 Arc Length . 148 3.2.2 Arc Length of Parametrically Defined Curves . 151 3.2.3 Tangents of Parametric Curves . 155 3.2.4 Areas Under Parametric Curves . 161 3.2.5 Summary . 164 3.3 Polar Coordinates . 165 3.3.1 Conversion Between Cartesian and Polar Coordinates 166 3.3.2 Polar Equations . 168 3.3.3 Tangents to Polar Curves . 171 3.3.4 Summary . 174 3.4 Areas and Lengths in Polar Coordinates . 174 3.4.1 Area . 174 3.4.2 Arc Length . 179 3.4.3 Summary . 180 3.5 Review . 181 Index 184 6 CONTENTS Chapter 1 Sequences and Series 1.1 Introduction This course is the third course in the calculus sequence, following MAT 167 and MAT 168. Its purpose is to prepare students for more advanced mathematics courses, particularly those pertaining to multivariable calculus (MAT 280) and numerical analysis (MAT 460 and 461). The course will focus on three main areas, which we briefly discuss here. 1.1.1 Sequences and Series Nearly all students have had to use a scientific calculator. Consider the fol- lowing: how does a calculator efficiently evaluate many of its functions, such as sin, cos or exp, when its hardware is only able to perform the four basic arithmetic operations, addition, subtraction, multiplication and division? The answer stems from the fact that it is generally not possible to eval- uate such functions exactly; rather, one has to settle for an approximation. However, this is no problem, because a calculator or computer can only rep- resent real numbers to limited accuracy anyway. To approximate a given function in a manner that is suitable for a calculator, we use infinite series, which is a sum of infinitely many terms. For example, we can write 1 1 1 X 1 exp x = 1 + x + x2 + x3 + ··· = xk: 2 6 k! k=0 The last term in the above equation uses sigma notation to express a sum of infinitely many terms in a concise way, when those terms can be described by a pattern. The above series is called a power series because each of its 7 8 CHAPTER 1. SEQUENCES AND SERIES terms features a power of x. When we study infinite series, we will consider important questions such as • When does an infinite series sum, or converge, to a finite number? We will see that in many cases, a sum of infinitely many terms does not converge at all, but rather continues growing, or diverging. For example, consider the two series 1 1 X 1 X 1 ; : n n1:01 n=1 n=1 Although the individual terms in the series may not differ by very much, especially for larger values of n, the first series does not con- verge, while the second one does. • If we truncate a series after a given number of terms, how well does the sum of the retained terms approximate the sum of the entire series? This is particularly important when using series to evaluate functions such as those implemented in scientific calculators. For example, sup- pose we take the abovementioned series for exp x and compute only the first 20 terms. If x = 0:1, then the result is correct to at least 16 significant digits. However, if x = 10, then we only obtain two correct digits. 1.1.2 Vectors and the Geometry of Space Next, we will become acquainted with three-dimensional space, in order to prepare you for later coursework in multivariable calculus. Our basic tools will be vectors, which can be used to represent either a position or direction in space. For example, if we represent three-dimensional space using Cartesian coordinates x, y and z, then the origin is the point with coordinates x = 0, y = 0 and z = 0, typically denoted by the ordered triple (0; 0; 0). Then, the point in space (1; 0; −2) has coordinates x = 1, y = 0 and z = −2, meaning that it is located 1 unit from the origin along the positive x-axis, 0 units from the origin along the y-axis, and 2 units from the origin along the negative z-axis. This is illustrated in Figure 1.1. We will use vectors to facilitate the description of, and operations on, lines and planes. This is particularly useful in computer graphics. Consider the problem of rendering a two-dimensional image, say for a frame in a film, of a collection of three-dimensional objects. To what point in 2-D space does any given point in 3-D space correspond? This question is answered by computing 1.1. INTRODUCTION 9 Figure 1.1: The origin in three-dimensional space, and the point (1; 0; −2). the projection of points in 3-D space onto a given plane in 2-D space that corresponds to the \screen". We will learn about an operation called the dot product that is used to compute projections. Another operation, called the cross product, is very useful for describing planes. 1.1.3 Parametric Equations and Polar Coordinates Finally, will study curves and surfaces in space. Often, it is necessary to describe the position of an object as a function of time. Therefore, we will study curves that are described by parametric equations, where the parameter is usually time. For example, in combat, the military needs to track the position of enemy projectiles over time. Using radar data, they can then construct parametric equations for the projectile's position in 3-D space, and then differentiate the equations with respect to time in order to estimate its velocity and then its trajectory, so that it can be intercepted. While we will work with Cartesian coordinates x, y and z for most of the course, we will find that it is often useful to represent curves or functions in the xy-plane using polar coordinates r and θ, where r represents distance from the origin, and θ represents the angle that the point makes with the origin and the positive x-axis.
Load more

Recommended publications
  • The Divergence As the Rate of Change in Area Or Volume
    The divergence as the rate of change in area or volume Here we give a very brief sketch of the following fact, which should be familiar to you from multivariable calculus: If a region D of the plane moves with velocity given by the vector field V (x; y) = (f(x; y); g(x; y)); then the instantaneous rate of change in the area of D is the double integral of the divergence of V over D: ZZ ZZ rate of change in area = r · V dA = fx + gy dA: D D (An analogous result is true in three dimensions, with volume replacing area.) To see why this should be so, imagine that we have cut D up into a lot of little pieces Pi, each of which is rectangular with width Wi and height Hi, so that the area of Pi is Ai ≡ HiWi. If we follow the vector field for a short time ∆t, each piece Pi should be “roughly rectangular”. Its width would be changed by the the amount of horizontal stretching that is induced by the vector field, which is difference in the horizontal displacements of its two sides. Thisin turn is approximated by the product of three terms: the rate of change in the horizontal @f component of velocity per unit of displacement ( @x ), the horizontal distance across the box (Wi), and the time step ∆t. Thus ! @f ∆W ≈ W ∆t: i @x i See the figure below; the partial derivative measures how quickly f changes as you move horizontally, and Wi is the horizontal distance across along Pi, so the product fxWi measures the difference in the horizontal components of V at the two ends of Wi.
    [Show full text]
  • Multivariable Calculus Workbook Developed By: Jerry Morris, Sonoma State University
    Multivariable Calculus Workbook Developed by: Jerry Morris, Sonoma State University A Companion to Multivariable Calculus McCallum, Hughes-Hallett, et. al. c Wiley, 2007 Note to Students: (Please Read) This workbook contains examples and exercises that will be referred to regularly during class. Please purchase or print out the rest of the workbook before our next class and bring it to class with you every day. 1. To Purchase the Workbook. Go to the Sonoma State University campus bookstore, where the workbook is available for purchase. The copying charge will probably be between $10.00 and $20.00. 2. To Print Out the Workbook. Go to the Canvas page for our course and click on the link \Math 261 Workbook", which will open the file containing the workbook as a .pdf file. BE FOREWARNED THAT THERE ARE LOTS OF PICTURES AND MATH FONTS IN THE WORKBOOK, SO SOME PRINTERS MAY NOT ACCURATELY PRINT PORTIONS OF THE WORKBOOK. If you do choose to try to print it, please leave yourself enough time to purchase the workbook before our next class in case your printing attempt is unsuccessful. 2 Sonoma State University Table of Contents Course Introduction and Expectations .............................................................3 Preliminary Review Problems ........................................................................4 Chapter 12 { Functions of Several Variables Section 12.1-12.3 { Functions, Graphs, and Contours . 5 Section 12.4 { Linear Functions (Planes) . 11 Section 12.5 { Functions of Three Variables . 14 Chapter 13 { Vectors Section 13.1 & 13.2 { Vectors . 18 Section 13.3 & 13.4 { Dot and Cross Products . 21 Chapter 14 { Derivatives of Multivariable Functions Section 14.1 & 14.2 { Partial Derivatives .
    [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]
  • 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]
  • MULTIVARIABLE CALCULUS Sample Midterm Problems October 1, 2009 INSTRUCTOR: Anar Akhmedov
    MULTIVARIABLE CALCULUS Sample Midterm Problems October 1, 2009 INSTRUCTOR: Anar Akhmedov 1. Let P (1, 0, 3), Q(0, 2, 4) and R(4, 1, 6) be points. − − − (a) Find the equation of the plane through the points P , Q and R. (b) Find the area of the triangle with vertices P , Q and R. Solution: The vector P~ Q P~ R = < 1, 2, 1 > < 3, 1, 9 > = < 17, 6, 5 > is the normal vector of this plane,× so equation− of− the− plane× is 17(x 1)+6(y −0)+5(z + 3) = 0, which simplifies to 17x 6y 5z = 32. − − − − − Area = 1 P~ Q P~ R = 1 < 1, 2, 1 > < 3, 1, 9 > = √350 2 | × | 2 | − − − × | 2 2. Let f(x, y)=(x y)3 +2xy + x2 y. Find the linear approximation L(x, y) near the point (1, 2). − − 2 2 2 2 Solution: fx = 3x 6xy +3y +2y +2x and fy = 3x +6xy 3y +2x 1, so − − − − fx(1, 2) = 9 and fy(1, 2) = 2. Then the linear approximation of f at (1, 2) is given by − L(x, y)= f(1, 2)+ fx(1, 2)(x 1) + fy(1, 2)(y 2)=2+9(x 1)+( 2)(y 2). − − − − − 3. Find the distance between the parallel planes x +2y z = 1 and 3x +6y 3z = 3. − − − Use the following formula to find the distance between the given parallel planes ax0+by0+cz0+d D = | 2 2 2 | . Use a point from the second plane (for example (1, 0, 0)) as (x ,y , z ) √a +b +c 0 0 0 and the coefficents from the first plane a = 1, b = 2, c = 1, and d = 1.
    [Show full text]
  • Math 56A: Introduction to Stochastic Processes and Models
    Math 56a: Introduction to Stochastic Processes and Models Kiyoshi Igusa, Mathematics August 31, 2006 A stochastic process is a random process which evolves with time. The basic model is the Markov chain. This is a set of “states” together with transition probabilities from one state to another. For example, in simple epidemic models there are only two states: S = “susceptible” and I = “infected.” The probability of going from S to I increases with the size of I. In the simplest model The S → I probability is proportional to I, the I → S probability is constant and time is discrete (for example, events happen only once per day). In the corresponding deterministic model we would have a first order recursion. In a continuous time Markov chain, transition events can occur at any time with a certain prob- ability density. The corresponding deterministic model is a first order differential equation. This includes the “general stochastic epidemic.” The number of states in a Markov chain is either finite or countably infinite. When the collection of states becomes a continuum, e.g., the price of a stock option, we no longer have a “Markov chain.” We have a more general stochastic process. Under very general conditions we obtain a Wiener process, also known as Brownian motion. The mathematics of hedging implies that stock options should be priced as if they are exactly given by this process. Ito’s formula explains how to calculate (or try to calculate) stochastic integrals which give the long term expected values for a Wiener process. This course will be a theoretical mathematics course.
    [Show full text]
  • Stokes' Theorem
    V13.3 Stokes’ Theorem 3. Proof of Stokes’ Theorem. We will prove Stokes’ theorem for a vector field of the form P (x, y, z) k . That is, we will show, with the usual notations, (3) P (x, y, z) dz = curl (P k ) · n dS . � C � �S We assume S is given as the graph of z = f(x, y) over a region R of the xy-plane; we let C be the boundary of S, and C ′ the boundary of R. We take n on S to be pointing generally upwards, so that |n · k | = n · k . To prove (3), we turn the left side into a line integral around C ′, and the right side into a double integral over R, both in the xy-plane. Then we show that these two integrals are equal by Green’s theorem. To calculate the line integrals around C and C ′, we parametrize these curves. Let ′ C : x = x(t), y = y(t), t0 ≤ t ≤ t1 be a parametrization of the curve C ′ in the xy-plane; then C : x = x(t), y = y(t), z = f(x(t), y(t)), t0 ≤ t ≤ t1 gives a corresponding parametrization of the space curve C lying over it, since C lies on the surface z = f(x, y). Attacking the line integral first, we claim that (4) P (x, y, z) dz = P (x, y, f(x, y))(fxdx + fydy) . � C � C′ This looks reasonable purely formally, since we get the right side by substituting into the left side the expressions for z and dz in terms of x and y: z = f(x, y), dz = fxdx + fydy.
    [Show full text]
  • Generalized Stokes' Theorem
    Chapter 4 Generalized Stokes’ Theorem “It is very difficult for us, placed as we have been from earliest childhood in a condition of training, to say what would have been our feelings had such training never taken place.” Sir George Stokes, 1st Baronet 4.1. Manifolds with Boundary We have seen in the Chapter 3 that Green’s, Stokes’ and Divergence Theorem in Multivariable Calculus can be unified together using the language of differential forms. In this chapter, we will generalize Stokes’ Theorem to higher dimensional and abstract manifolds. These classic theorems and their generalizations concern about an integral over a manifold with an integral over its boundary. In this section, we will first rigorously define the notion of a boundary for abstract manifolds. Heuristically, an interior point of a manifold locally looks like a ball in Euclidean space, whereas a boundary point locally looks like an upper-half space. n 4.1.1. Smooth Functions on Upper-Half Spaces. From now on, we denote R+ := n n f(u1, ... , un) 2 R : un ≥ 0g which is the upper-half space of R . Under the subspace n n n topology, we say a subset V ⊂ R+ is open in R+ if there exists a set Ve ⊂ R open in n n n R such that V = Ve \ R+. It is intuitively clear that if V ⊂ R+ is disjoint from the n n n subspace fun = 0g of R , then V is open in R+ if and only if V is open in R . n n Now consider a set V ⊂ R+ which is open in R+ and that V \ fun = 0g 6= Æ.
    [Show full text]
  • Calculus of Variations
    Calculus of Variations The biggest step from derivatives with one variable to derivatives with many variables is from one to two. After that, going from two to three was just more algebra and more complicated pictures. Now the step will be from a finite number of variables to an infinite number. That will require a new set of tools, yet in many ways the techniques are not very different from those you know. If you've never read chapter 19 of volume II of the Feynman Lectures in Physics, now would be a good time. It's a classic introduction to the area. For a deeper look at the subject, pick up MacCluer's book referred to in the Bibliography at the beginning of this book. 16.1 Examples What line provides the shortest distance between two points? A straight line of course, no surprise there. But not so fast, with a few twists on the question the result won't be nearly as obvious. How do I measure the length of a curved (or even straight) line? Typically with a ruler. For the curved line I have to do successive approximations, breaking the curve into small pieces and adding the finite number of lengths, eventually taking a limit to express the answer as an integral. Even with a straight line I will do the same thing if my ruler isn't long enough. Put this in terms of how you do the measurement: Go to a local store and purchase a ruler. It's made out of some real material, say brass.
    [Show full text]
  • May 1, 2019 LECTURE 25
    May 1, 2019 LECTURE 25: DIFFERENTIAL FORMS. 110.211 HONORS MULTIVARIABLE CALCULUS PROFESSOR RICHARD BROWN Synopsis. A continuation of the last three lectures on differential forms and their structure. P n 25.1. More notation. For ! = Fi1i2···im dxi1 ^· · ·^ dxim a differential m-form on M ⊂ R , n ≥ m, X ! = ··· Fi1i2···im dxi1 ^ · · · ^ dxim ; ˆM ˆ ˆM | {z } n-integrals where M is an m-dimensional region in Rn. Note that the order of the form and the dimension of the region integrated over will agree. Definition 25.1. Let f : D ⊂ Rn ! R be a C1-function. Then the exterior derivative of f, denoted df, is the 1-form @f @f df = dx1 + ::: + dxn = Df(x) dx = rf • dx: @x1 @xn P For ! = Fi1i2···im dxi1 ^ · · · ^ dxim a differential m-form, the differential (m + 1)-form X d! = d(Fi1i2···im ) ^ dxi1 ^ · · · ^ dxim is called the exterior derivative of !. Some notes: • We call a C1-function f : D ⊂ Rn ! R a (differential) 0-form. Thus the exterior derivative of a function is simply its differential, a 1-form. Thus theexterior derivative of any differential m-form is always an (m + 1)-form. • For each set of indices, the term d(Fi1i2···im ) is the standard differential of a function, and is a 1-form. Upon writing it out, one must then address any and all simplifications and cancellations, which can be many. Example 25.1. Let ! = x2y dx − x dy be a C1 1-form on R2. Then d! = d(x2y) ^ dx − d(x) ^ dy = (2xy dx + x2 dy) ^ dx − (1 dx − 0 dy) ^ dy = 2xy dx ^ dx + x2 dy ^ dx − dx ^ dy = −(1 + x2) dx ^ dy: So what is d(d!) = d2!? (Hint: Is it possible to have a 3-form on the plane?) Here d(d!) = d(−(1 + x2)) dx ^ dy = −2x dx ^ dx ^ dy = 0: (Why?) 1 2 110.211 HONORS MULTIVARIABLE CALCULUS PROFESSOR RICHARD BROWN Example 25.2.
    [Show full text]
  • The Calculus of Variations
    The Calculus of Variations Jeff Calder January 19, 2017 Contents 1 Introduction2 1.1 Examples.......................................2 2 The Euler-Lagrange equation6 2.1 The gradient interpretation............................. 10 3 Examples continued 12 3.1 Shortest path..................................... 12 3.2 The brachistochrone problem............................ 13 3.3 Minimal surfaces................................... 16 3.4 Minimal surface of revolution............................ 18 3.5 Image restoration................................... 22 3.6 Image segmentation................................. 27 4 The Lagrange multiplier 30 4.1 Isoperimetric inequality............................... 31 5 Sufficient conditions 34 5.1 Basic theory of convex functions.......................... 35 5.2 Convexity is a sufficient condition.......................... 37 A Mathematical preliminaries 38 A.1 Integration...................................... 38 A.2 Inequalities...................................... 39 A.3 Partial derivatives.................................. 40 A.4 Rules for differentiation............................... 41 A.5 Taylor series...................................... 43 A.5.1 One dimension................................ 43 A.5.2 Higher dimensions.............................. 44 A.6 Topology....................................... 47 A.7 Function spaces.................................... 48 A.8 Integration by parts................................. 50 A.9 Vanishing lemma................................... 51 A.10 Total variation
    [Show full text]
  • Corral's Vector Calculus
    1 0.8 0.6 0.4 0.2 z 0 -0.2 -10 -0.4 -5 -10 0 -5 x 0 5 5 y 10 10 CORRAL’S VECTOR CALCULUS Michael Corral and Anton Petrunin Corral’s Vector Calculus Michael Corral and Anton Petrunin About the author: Michael Corral is an Adjunct Faculty member of the Department of Mathematics at Schoolcraft College. He received a B.A. in Mathematics from the University of California at Berkeley, and received an M.A. in Mathematics and an M.S. in Industrial & Operations Engineering from the University of Michigan. This text was typeset in LATEX2ε with the KOMA-Script bundle, using the GNU Emacs text editor on a Fedora Linux system. The graphics were created using MetaPost, PGF, and Gnuplot. Copyright ©2016 Anton Petrunin. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled “GNU Free Documentation License”. Preface This book covers calculus in two and three variables. It is suitable for a one-semester course, normally known as “Vector Calculus”, “Multivariable Calculus”, or simply “Calculus III”. The prerequisites are the standard courses in single-variable calculus (also known as Cal- culus I and II). The exercises are divided into three categories: A, B and C. The A exercises are mostly of a routine computational nature, the B exercises are slightly more involved, and the C exercises usually require some effort or insight to solve.
    [Show full text]