MULTIVARIABLE CALCULUS in PROBABILITY Math21a, O. Knill
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Foundations of Probability Theory and Statistical Mechanics
Chapter 6 Foundations of Probability Theory and Statistical Mechanics EDWIN T. JAYNES Department of Physics, Washington University St. Louis, Missouri 1. What Makes Theories Grow? Scientific theories are invented and cared for by people; and so have the properties of any other human institution - vigorous growth when all the factors are right; stagnation, decadence, and even retrograde progress when they are not. And the factors that determine which it will be are seldom the ones (such as the state of experimental or mathe matical techniques) that one might at first expect. Among factors that have seemed, historically, to be more important are practical considera tions, accidents of birth or personality of individual people; and above all, the general philosophical climate in which the scientist lives, which determines whether efforts in a certain direction will be approved or deprecated by the scientific community as a whole. However much the" pure" scientist may deplore it, the fact remains that military or engineering applications of science have, over and over again, provided the impetus without which a field would have remained stagnant. We know, for example, that ARCHIMEDES' work in mechanics was at the forefront of efforts to defend Syracuse against the Romans; and that RUMFORD'S experiments which led eventually to the first law of thermodynamics were performed in the course of boring cannon. The development of microwave theory and techniques during World War II, and the present high level of activity in plasma physics are more recent examples of this kind of interaction; and it is clear that the past decade of unprecedented advances in solid-state physics is not entirely unrelated to commercial applications, particularly in electronics. -
Randomized Algorithms Week 1: Probability Concepts
600.664: Randomized Algorithms Professor: Rao Kosaraju Johns Hopkins University Scribe: Your Name Randomized Algorithms Week 1: Probability Concepts Rao Kosaraju 1.1 Motivation for Randomized Algorithms In a randomized algorithm you can toss a fair coin as a step of computation. Alternatively a bit taking values of 0 and 1 with equal probabilities can be chosen in a single step. More generally, in a single step an element out of n elements can be chosen with equal probalities (uniformly at random). In such a setting, our goal can be to design an algorithm that minimizes run time. Randomized algorithms are often advantageous for several reasons. They may be faster than deterministic algorithms. They may also be simpler. In addition, there are problems that can be solved with randomization for which we cannot design efficient deterministic algorithms directly. In some of those situations, we can design attractive deterministic algoirthms by first designing randomized algorithms and then converting them into deter- ministic algorithms by applying standard derandomization techniques. Deterministic Algorithm: Worst case running time, i.e. the number of steps the algo- rithm takes on the worst input of length n. Randomized Algorithm: 1) Expected number of steps the algorithm takes on the worst input of length n. 2) w.h.p. bounds. As an example of a randomized algorithm, we now present the classic quicksort algorithm and derive its performance later on in the chapter. We are given a set S of n distinct elements and we want to sort them. Below is the randomized quicksort algorithm. Algorithm RandQuickSort(S = fa1; a2; ··· ; ang If jSj ≤ 1 then output S; else: Choose a pivot element ai uniformly at random (u.a.r.) from S Split the set S into two subsets S1 = fajjaj < aig and S2 = fajjaj > aig by comparing each aj with the chosen ai Recurse on sets S1 and S2 Output the sorted set S1 then ai and then sorted S2. -
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. -
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 . -
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. -
Probability Theory Review for Machine Learning
Probability Theory Review for Machine Learning Samuel Ieong November 6, 2006 1 Basic Concepts Broadly speaking, probability theory is the mathematical study of uncertainty. It plays a central role in machine learning, as the design of learning algorithms often relies on proba- bilistic assumption of the data. This set of notes attempts to cover some basic probability theory that serves as a background for the class. 1.1 Probability Space When we speak about probability, we often refer to the probability of an event of uncertain nature taking place. For example, we speak about the probability of rain next Tuesday. Therefore, in order to discuss probability theory formally, we must first clarify what the possible events are to which we would like to attach probability. Formally, a probability space is defined by the triple (Ω, F,P ), where • Ω is the space of possible outcomes (or outcome space), • F ⊆ 2Ω (the power set of Ω) is the space of (measurable) events (or event space), • P is the probability measure (or probability distribution) that maps an event E ∈ F to a real value between 0 and 1 (think of P as a function). Given the outcome space Ω, there is some restrictions as to what subset of 2Ω can be considered an event space F: • The trivial event Ω and the empty event ∅ is in F. • The event space F is closed under (countable) union, i.e., if α, β ∈ F, then α ∪ β ∈ F. • The even space F is closed under complement, i.e., if α ∈ F, then (Ω \ α) ∈ F. -
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 -
Effective Theory of Levy and Feller Processes
The Pennsylvania State University The Graduate School Eberly College of Science EFFECTIVE THEORY OF LEVY AND FELLER PROCESSES A Dissertation in Mathematics by Adrian Maler © 2015 Adrian Maler Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December 2015 The dissertation of Adrian Maler was reviewed and approved* by the following: Stephen G. Simpson Professor of Mathematics Dissertation Adviser Chair of Committee Jan Reimann Assistant Professor of Mathematics Manfred Denker Visiting Professor of Mathematics Bharath Sriperumbudur Assistant Professor of Statistics Yuxi Zheng Francis R. Pentz and Helen M. Pentz Professor of Science Head of the Department of Mathematics *Signatures are on file in the Graduate School. ii ABSTRACT We develop a computational framework for the study of continuous-time stochastic processes with c`adl`agsample paths, then effectivize important results from the classical theory of L´evyand Feller processes. Probability theory (including stochastic processes) is based on measure. In Chapter 2, we review computable measure theory, and, as an application to probability, effectivize the Skorokhod representation theorem. C`adl`ag(right-continuous, left-limited) functions, representing possible sample paths of a stochastic process, form a metric space called Skorokhod space. In Chapter 3, we show that Skorokhod space is a computable metric space, and establish fundamental computable properties of this space. In Chapter 4, we develop an effective theory of L´evyprocesses. L´evy processes are known to have c`adl`agmodifications, and we show that such a modification is computable from a suitable representation of the process. We also show that the L´evy-It^odecomposition is computable. -
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. -
Curriculum Vitae
Curriculum Vitae General First name: Khudoyberdiyev Surname: Abror Sex: Male Date of birth: 1985, September 19 Place of birth: Tashkent, Uzbekistan Nationality: Uzbek Citizenship: Uzbekistan Marital status: married, two children. Official address: Department of Algebra and Functional Analysis, National University of Uzbekistan, 4, Talabalar Street, Tashkent, 100174, Uzbekistan. Phone: 998-71-227-12-24, Fax: 998-71-246-02-24 Private address: 5-14, Lutfiy Street, Tashkent city, Uzbekistan, Phone: 998-97-422-00-89 (mobile), E-mail: [email protected] Education: 2008-2010 Institute of Mathematics and Information Technologies of Uzbek Academy of Sciences, Uzbekistan, PHD student; 2005-2007 National University of Uzbekistan, master student; 2001-2005 National University of Uzbekistan, bachelor student. Languages: English (good), Russian (good), Uzbek (native), 1 Academic Degrees: Doctor of Sciences in physics and mathematics 28.04.2016, Tashkent, Uzbekistan, Dissertation title: Structural theory of finite - dimensional complex Leibniz algebras and classification of nilpotent Leibniz superalgebras. Scientific adviser: Prof. Sh.A. Ayupov; Doctor of Philosophy in physics and mathematics (PhD) 28.10.2010, Tashkent, Uzbekistan, Dissertation title: The classification some nilpotent finite dimensional graded complex Leibniz algebras. Supervisor: Prof. Sh.A. Ayupov; Master of Science, National University of Uzbekistan, 05.06.2007, Tashkent, Uzbekistan, Dissertation title: The classification filiform Leibniz superalgebras with nilindex n+m. Supervisor: Prof. Sh.A. Ayupov; Bachelor in Mathematics, National University of Uzbekistan, 20.06.2005, Tashkent, Uzbekistan, Dissertation title: On the classification of low dimensional Zinbiel algebras. Supervisor: Prof. I.S. Rakhimov. Professional occupation: June of 2017 - up to Professor, department Algebra and Functional Analysis, present time National University of Uzbekistan. -
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. -
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.