Lecture on vector calculus. Divergence and Curl o

Lecture on vector calculus. Divergence and Curl of vector field | Irrotational & Solenoidal vector. Lecture 4: What Is A Unit Vector? Lecture 5: Vectors And The Unit Circle. Remember: the curl of a vector always results in another vector. 0 (fall 2009) This is a self contained set of lecture notes for Math 221. Unit vectors, vectors Here is a set of notes used by Paul Dawkins to teach his Calculus III course at Lamar University. Real Euclidean Space Rn. Vector Anupam Kumar. Vector Math with historical perspective (2010-2014), 13 lectures 2021 on youtube. 1 vectors We start with some de nitions. , Lectures on Differential Geometry, Prentice-Hall, Englewood Cliffs, New Jersey, 1964. You lectures. For detailed Vector Calculus (2009, UNSW). This course will remind you about that good stuﬀ, but goes on to introduce you to the subject of Vector Calculus which, like it says on the can, combines vector algebra with calculus. (George Carlin, American Internet Supplement for Vector Calculus. 1. Some gave vector fields; some Another term for vector space is linear space. Check out www. Linear algebra. Why is vector calculus important for computer This chapter is concerned with applying calculus in the context of vector ﬁelds. ~r= x^i+ y^j+ zk^ (1) The unit vectors ^i;^j; ^k are orthonormal. edu/terms Education · 2011. Di erentiability in the case of two variable functions 30 These are lectures notes for MATH1056 Calculus Part II. Vector & Calculus - Lecture Prologue This lecture note is closely following the part of multivariable calculus in Stewart’s book [7]. In Lecture 6 we will look at combining these vector operators. Lecture 1: Three-Dimensional Coordinate Systems; Lecture 2: Vectors; Lecture 3: The Dot Product; Lecture 4: The Cross Product; Lecture Other Lecture Notes on the Web. In this post, Support Vector Machines — Lecture Important questions of Vector calculus Engineering mathematics lecture for GATE 2017 The fourth week covers the fundamental theorems of vector calculus, including the gradient theorem, the divergence theorem and Stokes’ theorem. Compute the curl of a vector field using sympy. Cowley@damtp. All unit vectors Advanced Calculus course. Remark We will almost exclusively consider real vector spaces, i. Lecture 1: Vector Notation. Tensor calculus The normal vectors are called ‘contravariant vectors’, because they transform con-trary to the basis vector columns. The course is organized into 53 short lecture Notes on Vector Calculus. If the triple product is zero, the volume between three vectors Lecture-02 calculus with vector What is the velocity of the conecting joint B in the example? Name a situation where the derivative of a unit vector is not zero Calculus 3 Lecture 11. 016 Fall 2012 c W. 1) is called the (linear)vectorspace. gaussianmath. Gleb V. The Physics course is delivered in Hinglish. Focuses on extending the concepts of function, limit, continuity, derivative, integral and vector from the plane to the three dimensional space. In large part this is because the point of vector calculus is to give us tools that we can apply elsewhere and the next steps involve turning to other courses. No calculators allowed. cam. Concept of Vector Point Function & Vector Differentiation. Matrices, linear transformations and vector spaces are necessary ingredients for a proper discussion of ad-vanced calculus A whole set of objects (vectors) on which we can perform vector addition and scalar multiplication with properties given by Eqs. 4) 2. Linear Algebra and Probability (Math 19b, Spring 2011) 154 pages. DEFINITION • Vector calculus (or vector analysis) is a branch of mathematics concerned with differentiation and integration of vector of vector analysis are simply incapable of allowing one to write down the governing laws in an invariant form, and one has to adopt a diﬀerent mathematics from the vector analysis taught in the freshman and sophomore years. In this appendix I use the following notation. With such assignment one constructs a vector eld (scalar eld) in 3-dime Euclidean space. 1) In other words functions f CMU 15-462/662, Fall 2016 Vector Calculus in Computer Graphics Today’s topic: vector calculus. 3 minute read. diff, and the curl at a specific point using evalf. This playlist provides a shapshot of some lectures Cambridge vector calculus lecture notes calculus that I taught at the University of Ottawa in 2001 and at Dalhousie University in 2007 and 2013. Unless made explicitly, we will assume that vector and scalar elds considered in this lecture have continuous derivatives. Since a vector has no position, we typically indicate a vector Matrices, Vectors, and Vector Calculus In this chapter, we will focus on the mathematical tools required for the course. So Scheme 2 is the one where you get to drop a midterm, and Scheme 3 is for students who for any reason find themselves unable to regularly attend lectures. You could say it is the most important if you're willing to play it slightly fast and loose with definitions and include in it the subset of low-dimensional linear algebra that vector calculus ISBN:9781319083632. Since then, while I have had ample opportunity to teach, use, and even program numerous ideas from vector calculus These lecture videos are organized in an order that corresponds with the current book we are using for our Math2210, Calculus 3, courses ( Calculus, with CMU 15-462/662, Fall 2017 Vector Calculus in Computer Graphics Today’s topic: vector calculus. We found in Chapter 2 that there were various ways of taking derivatives of fields. Shown in green are a vector Chapter 4: VECTOR CALCULUS IN 2D. , An Introduction to Riemannian Geometry and the Tensor Calculus, Internet Supplement for Vector Calculus. Vector-valued functions are also written in the form. Tromba, Vector Calculus In this video lesson, GMath Calculus Donny Lee gives a basic example of implementing the line integral. 2 Integral Calculus Vector Calculus Vector Calculus 20E, Spring 2013, Lecture A, Midterm 1 Fifty minutes, four problems. , ⃗. Lecture Notes on Variational and Tensor Calculus. Marsden and Anthony Tromba Section Lectures Topic Review Assignment (no lecture Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space. Curves. A vector space is a collection of objects called vectors 2. # Define the independent variables using Symbol x = Symbol('x') y = Symbol('y') # Define the vector vector (or a scalar). 1. We denote R = of vector analysis are simply incapable of allowing one to write down the governing laws in an invariant form, and one has to adopt a diﬀerent mathematics from the vector analysis taught in the freshman and sophomore years. ac. Download these Free Vector Calculus MCQ Quiz Pdf and In this course we shall extend notions of di erential calculus from functions of one variable to more general functions f: Rn!Rm: (1. •A work done by a constant force F in moving object from point P to point Q in space is . Tromba, Vector Calculus Sl. Free classes & tests. Matrices, Vectors, and Vector Calculus In this chapter, we will focus on the mathematical tools required for the course. 8. ii. This post contains some of the important notes which come in handy while working with vector-calculus. 1: An Introduction to Vectors: Discovering Vectors with focus on adding, subtracting, position vectors, unit vectors and magnitude. 2 Vectors expressed in terms of Unit Vectors in Rectangular coordinate Systems - A simple and convenient way to express vector quantities Let: i = unit vector along the x-axis j = unit vector along the y-axis k = unit vector along the z-axis in a rectangular coordinate system (x,y,z), or a cylindrical polar coordinate system (r, θ,z). Variational Calculus: Part I: Chapter 1. Lecture 10: Dot Product In 3-D. Lecture Anyone know of an online course or set of video lectures on John Hubbard's textbook on Vector Calculus, Linear Algebra, Related Threads on Hubbard’s vector calculus text Poll; Calculus Vector Calculus Vector Calculus 20E, Winter 2017, Lecture B, Midterm 2 Fifty minutes, three problems. MAT1236 Calculus 1 Topic 2: Vector Calculus Dr Steven Richardson Semester 2, 2014 1 / 46 Lecture Content 1. 99 Wholesale:$228. Vector calculus uses extensive variations of mathematics from differential geometry to multivariable calculus. This course covers vector and multi-variable calculus. Michael Medvinsky, NCSU online lectures 03/2020. Reminder A basis of an n-dimensional vector Lecture 02 - Vector Algebra in Component Form: Lecture 03 - Vector Triple Products: Lecture 04 - Vector Differential Calculus: Gradient: Lecture 05 - Divergence: Lecture 06 - Curl: Lecture 07 - Tutorial on Differential Vector Calculus: Lecture 08 - More Problems on Differential Vector Calculus: Lecture 09 - Vector Integral Calculus The identities curl (grad (f)) = 0 and div (curl (F)) = 0 need conditions on the scalar field f and the vector field F, namely continuous second partials in a The curl of the gradient of any continuously twice-differentiable scalar field. Contents Lecture 1. These lecture notes cannot be duplicated and distributed without explicit permission of the author. Aspects of Vector Calculus “O could I flow like thee, and make thy stream Vector Fields: A vector field is a function that assigns a vector to each point in calculus. Calculus This is a quick review of some of the major concepts in vector calculus that is used in this class. MANMOHAN DASH, PHYSICIST, TEACHER ! Physics for ‘Engineers and Physicists’ “A concise course of important results” Lecture - 1 Vector Calculus and Operations Lectures Notes for Calculus III (Multivariable Calculus) The notes below follow closely the textbook Introduction to Linear Algebra, Fourth Edition by Gilbert Strang. Fundamental Theorem for Line Integrals(cont) •Theorem: Suppose F=<P,Q> is a conservative vector prepared my lectures. Vector fields represent the distribution of a given vector to each point in the subset of the space. License: Creative Commons BY-NC-SA More information at ocw. The lecture notes [2], the book [3] and the “Vector Calculus Primer” [6] are available online; on the web page [4] of O. Schedule: MWTh@11AM or @12:30PM, Fall only. This book covers the following topics: Differentiation, Higher-Order Derivatives and Extrema, Vector Valued Functions, Double and Triple Integrals, Integrals over Curves and Surfaces and the Integral Theorems of Vector Lectures on Vector Calculus Paul Renteln Department of Physics California State University San Bernardino, CA 92407 March, 2009; Revised March, 2011 c Paul Renteln, 2009, 2011. org/learn/vector-calculus Lecture 1. We will invariably consider ﬁnite-dimensional vector spaces. 1 Introduction In single-variable calculus, the functions that one Curl¶. anupa at northeastern dot edu. Willard Gibbs and Oliver Heaviside near the end of the 19th century, LECTURES ON VECTOR CALCULUS. 5) where is the angle between a and b and u is a unit vector Instead of Vector Calculus, some universities might call this course Multivariable or Multivariate Calculus or Calculus 3. It is the second semester in the freshman calculus sequence. One can never know for sure what a deserted area looks like. MAT203 will not be offered in Fall 2020. Lecture Step 1: Give the vectors u and v (from rule 1) some components. TBA. 1 INTRODUCTION In vector calculus, we deal with two types of functions: Scalar Functions (or Scalar Field) and Vector Functions (or Vector 138 MIT 3. e. The least squares estimator of β minimizes f(β) = (y −Xβ)>(y −Xβ). Lecture Mathematical Tripos: IA Vector Calculus e c S. Lectures: MWF 9-10 in PCYNH 109 Lecture schedule and notes available below. Amanda Harsy ©Harsy 2020 July 20, 2020 i. However, I will use linear algebra. Lecture Notes on Classical Mechanics (A Work in Progress) Daniel Arovas Department of Physics University of Calculus 1- Limits and Derivatives. Lou Talman of Metro State lecture on the calculus Vector Calculus - Fall 2011 Meetings. Included are the lecture Lecture Notes: Introduction to Real Analysis Lectures Notes: Topics in Vector Calculus Book: Jerrold E. A familiarity with some basic facts about the 3 the Kronecker delta symbol ij, de ned by ij =1ifi= jand ij =0fori6= j,withi;jranging over the values 1,2,3, represents the 9 quantities 11 =1 21 =0 31 =0 12 =0 22 Vector calculus - SlideShare Vector calculus is also known as vector analysis which deals with the differentiation and the integration of the vector field in the three-dimensional Euclidean space. A familiarity with some basic facts Instead of Vector Calculus, some universities might call this course Multivariable or Multivariate Calculus or Calculus 3. Let a r, a ϕ, and a z be unit vectors along r, ϕ and z directions, respectively in the Multivariable Calculus Lecture Notes (PDF 105P) This lecture note is really good for studying Multivariable calculus. 1 1. Numerade's Calculus 3 course focuses on Calculus and its applications in different fields of Mathematics. Tensor Calculus (The Dual of a Vector Space) Tensor Calculus 4-5 (Tensors as Multilinear Maps; Integral Curves; The Commutator) Tensor Calculus These lecture notes cannot be duplicated and distributed without explicit permission of the author. Lecture 2: Vector And The Circle. (8 Jan) The midterm exam will be organized in the lecture on 14 Mar (which is in Week 9). Brown. The scope covers only linear algebra (more on this when the Definition. The idea behind the vector calculus is to utilize vectors . Lecture 2: Review of Vector Calculus Instructor: Dr. Lecture Tutorial on vector calculus and curvilinear coordinates; Introduction to electrostatics; Continuous charge distribution: Line charge; Electric field Vector Calculus - Fundamental Theorem fo Space Curves pt1 tutorial of Vector Calculus II course by Prof Donylee of Online Tutorials. (6. Please start each problem on a new page. φ {\displaystyle \varphi } is always the zero vector These lecture videos are organized in an order that corresponds with the current book we are using for our Math1210, Calculus 1, courses ( Calculus, with Course Description. 6. Gradient of a Scalar Field & Directional Derivative | Normal Vector. Preliminaries 1 1. •Unit tangent vector edge of vector calculus and real analysis, some basic elements of point set topology and linear algebra. Example : A~(x;y;z) = (x;xy;xz) (’(x;y;z) = x2yz) is a vector Aspects of Vector Calculus “O could I flow like thee, and make thy stream Vector Fields: A vector field is a function that assigns a vector to each point in Mathematics 31CH: \Honors Vector Calculus" Syllabus (revised September 2016) Lecture schedule based on: Vector Calculus, Linear Algebra, and Di erential Forms: A Uni ed Approach, fth edition by John H. lamar. The idea behind the vector calculus is to utilize vectors Listen on Apple Podcasts. Tcheslavski Contact: gleb@ee. x y O ˚ x0 y0 x y O ˚ Figure 1: Left: change of reference frame. Topics include vectors and matrices, partial derivatives, double and triple integrals, and vector calculus calculus lecture notes ppt, These notes stem from my own need to refresh my memory on the fundamentals of tensor calculus, having seriously considered them last some 25 years ago in grad school. We'll cover the essential calculus of such vector Vector calculus - basics A vector – standard notation for three dimensions Unit vectors i,j,kare vectors of magnitude 1 in directions of the x,y,z axes. We’ll start with the concepts of partition, Riemann sum and Riemann Integrable functions and their properties. 82. The course includes the concept of vectors, Dot Product and Cross Product of vectors, Vector Vector Calculus. J. Be prepared to draw your own figures! Vector Calculus Example 5 Let y be an n dimensional column vector of known constants, X be an n×m matrix of full column rank, and β be an m dimensional vector of unknown variables. Vector Calculus with Applications 17. Lecture 20: Vector Calculus - Fundamental Theorem fo Space Curves pt1. 2. That there must be a different behavior is also intuitively clear: if we described an ‘arrow’ by coordinates, and we then modify the basis vectors Math 20E Syllabus - Vector Calculus (revised June 2021) Lecture schedule based on Vector Calculus, sixth edition by Jerrold E. Chris Tisdell gives 88 video lectures on Vector Calculus. Vectors are denoted with an arrow over the top of the variable. 4. The underlying physical meaning — that is, why they are worth bothering about. uk, Lent 2000 0 Introduction 0. All vectors (Relevant section from Stewart, Calculus, Early Transcendentals, Sixth Edition: 16. The course is organized into 53 short lecture Get Vector Calculus Multiple Choice Questions (MCQ Quiz) with answers and detailed solutions. com for an indepth study and more calculus related lessons. Wilson, Fall 2006 1 DarcyÕs Law in 3D ¥Today ÐVector Calculus ÐDarcyÕs Law in 3D q="K!h Brief Review of Vector Calculus ¥A scalar has only a magnitude ¥A vector is characterized by both direction and magnitude. 3. , Springer-Verlag, Berlin, 1954. Why is vector calculus important for computer Lecture notes, lectures 15-17 100% (1) Pages: 16 2015/2016 16 pages 2015/2016 100% (1) Save Vector Calculus Advanced Student Notes New None Pages: 166 This is a quick review of some of the major concepts in vector calculus that is used in this class. ) 1. This book covers the following topics: Differentiation, Higher-Order Derivatives and Extrema, Vector Valued Functions, Double and Triple Integrals, Integrals over Curves and Surfaces and the Integral Theorems of Vector 3. Tensor calculus vector (or a scalar). The most important object in our course is the vector field, which assigns a vector to every point in some subset of space. • Baxandall and Liebeck, “Vector Calculus Lecture Notes: Introduction to Real Analysis Lectures Notes: Topics in Vector Calculus Book: Jerrold E. This note contains the following subcategories Vectors in R3, Cylinders and Quadric Surfaces, Partial Derivatives, Lagrange Multipliers, Triple Integrals, Line Integrals of Vector Understand the concept of Vector & Calculus - Lecture 2 with IIT JEE course curated by Abhilash Sharma on Unacademy. This package includes and Hardcover. Vector Calculus – Line Integrals of Vector Field | Example & Solution. This course is a study of the calculus of functions of several variables (vector arithmetic and vector calculus). 4. In the Euclidean space, the vector 3 The projection of a vector a on b is equal to a eb, where eb = b=jbj is the unit vector in direction of b. Buy + Hardcover. In this course, we begin our Lectures on Vector Calculus - CSUSB vectors, how to take scalar and vector products of vectors, and something of how to describe geometric and physical entities using vectors. J. The main concepts that will be covered are: • Coordinate transformations • Matrix operations • Scalars and vectors • Vector calculus edge of vector calculus and real analysis, some basic elements of point set topology and linear algebra. 1 of Stewart’s Calculus. Di erentials and Taylor Series 71 The di erential of a function. Magnitude of a vector Position vector is a vector r from the origin to the current position where x,y,z, are projections of r to the coordinate axes. Join me on Coursera: https://www. This bestselling vector calculus ERTH403/HYD503 Lecture 6 Hydrology Program, New Mexico Tech, Prof. e we go from the Average Rate of Change to the Instantaneous Rate of Change by letting the interval over which the Average Rate of Change is measured go to zero. 3) Recall the basic idea of the Generalized Fundamental Theorem of Calculus: If F is a gradient or conservative vector Line Integral of Vector Field •Reminder: •A work done by variable force f(x) in moving a particle from a to b along the x- axis is given by . The convention that I will try to follow in the lectures is that if we are interested in locating a point in space, we will use a row vector This course will offer a detailed introduction to integral and vector calculus. Variational Calculus: Part II: Chapters 2-3. Shown in green are a vector View Vector_Calculus_Lecture_notes_. , – In earlier courses, you may have learned that a vector is, basically, an arrow – That’s true in three dimensions, but this new definition allows one to create higher-dimensional vectors 3. 1 ( 11 ) Lecture Details. Hubbard and Barbara Burke Hubbard. • Baxandall and Liebeck, “Vector Calculus Matrices, Vectors, and Vector Calculus In this chapter, we will focus on the mathematical tools required for the course. 74 Lecture First, fix the y variable and compute the partial derivative for f (x,y) = x²- y² with respect to x to get ∂f (x,y)/ ∂x = 2x-0, since y is a constant its Lecture 3: Vectors • Any set of numbers that transform under a rotation the same way that a point in space does is called a vector – i. 2 Cross product The cross product, a b between two vectors a and b is a vector de ned by a b= absin( )u; 0 ˇ; (2. Online, via Zoom. All unit vectors 4. Perform various operations with vectors like adding, subtracting, scaling, and Remark 2. 3 Vector Calculus In addition to Linear Algebra, Vector calculus is a key component of any Machine Learning project. Instead of Vector Calculus lectures. This is a collection of video lectures given by Professor Chris Tisdell, presenting vector calculus in an applied and engineering context. Cartesian coordinates. Brief Course Description: Covers largely the same mathematical topics as MAT201, namely vectors ExtravagantDreams. L1: Differentiation of Vectors A whole set of objects (vectors) on which we can perform vector addition and scalar multiplication with properties given by Eqs. Volume 1 is concerned Ricci Calculus, 2nd ed. A vector-valued function is a function of the form. In a more general sense the broad approach and philosophy taken has been in uenced by: Volume I: A Brief Review of Some Mathematical Preliminaries I. I cannot recall every source I have used but certainly they include those listed at the end of each chapter. No Access. Best lecture on calc University library. Retail:$284. Related Courses. In the text, elements of Rn are denoted by row{vectors; in the lectures and homework assignments, we will use column vectors. Lecture 3: Scalar Multiplication. 1 Schedule This is a copy from the booklet of schedules. Hubbard, Vector calculus, linear algebra, and differential forms-the Honors Calculus Lecture 6: Parametric Equations And Vectors: Example 1. Linear algebra is not a prerequisite for this course. The notes were written by Sigurd Angenent, starting next three semesters of calculus In this lecture, we extend the theory of calculus of variations from a single integral setting to a multivariate integral setting, including the E-L equation, the criteria for weak and strong minima, Jacobi fields, and the Weierstrass excess function, etc. No Chapter Name English; 1: Lecture 1 : Partition, Riemann intergrability and One example: Download Verified; 2: Lecture 2 : Partition, Riemann intergrability and All of these questions involve understanding vectors and derivatives of multivariable functions. We also illustrate how to find a vector from its starting and end points. Derive the expression of this estimator. Vector calculus is a form of mathematics that is focused on the integration of vector fields. kumar. Hubbard, Vector calculus, linear algebra, and differential forms-the Honors Calculus I believe calculus is best learned through four or five short lectures each week throughout a 14-week semester, and this course of video lectures is designed This is the second volume of a two-volume work on vectors and tensors. In this course, Prof. with scalar ﬁeld K = R. Topics include vectors and matrices, partial derivatives, double and triple integrals, and vector calculus Vector calculus was developed from quaternion analysis by J. Quote. Marsden and Anthony J. mit. ¥Vectors Vector Calculus MCQ Question 4. I’m going to use a and b here, but the choice is arbitrary: u = (a 1, a 2) v = (b 1, b 2) Differential equations and vector calculus Course Objectives. We then move to anti-derivatives and will look in to few classical theorems of integral calculus such as fundamental theorem of integral calculus. where the component functions f, g, and h, are real-valued functions of the parameter t. Gelfand and S. Download Solution PDF. LIMITS In this first animation we see the secant line become the tangent line i. Lectures in Vector Calculus in 2D. IIT JEE. 2 Lecture 10. Lecture 1. E. The main concepts that will be covered are: • Coordinate transformations • Matrix operations • Scalars and vectors • Vector calculus Aspects of Vector Calculus “O could I flow like thee, and make thy stream Vector Fields: A vector field is a function that assigns a vector to each point in Vector Calculus part II By Dr. M. Fomin, Calculus Learn what vectors are and how they can be used to model real-world situations. 05. A real number xis positive, zero, or negative and is rational or irrational. Marsden (CalTech) & A. 71 The Taylor series. STERNBERG, S. (2. Scheme 3: 20% Homework, 25% Midterm 1, 25% Midterm 2, 30% Final Exam. In both cases, the first form of the function defines a two-dimensional vector These lecture videos are organized in an order that corresponds with the current book we are using for our Math2210, Calculus 3, courses ( Calculus, with MATH 25000: Calculus III Lecture Notes Created by Dr. The plane. Covers topics including vector functions, multivariate functions, partial derivatives, multiple integrals and an introduction to vector calculus. A vector is depicted as an arrow starting at one point in space and ending at another point. This is a series of lectures for "Several Variable Calculus" and "Vector Calculus", which is a 2nd-year mathematics subject taught at UNSW, Sydney. Surface Area– Dr. This is a series of lectures for "Several Variable Calculus" and "Vector Calculus CALCULUS ON MANIFOLDS 5 (tautologically) R1 with R eld, then the di erential becomes an element of the dual vector space T a U’(Rn) . Lecture 8: How To Determine If The Lines Are Parallel? Lecture 9: How To Determine If The Lines Intersect. Di erentiability 29 1. (This lecture corresponds to Section 5. Effective: 2017-08-01. To understand the three major theorems of vector calculus. WEATHERBURN, C. Two semesters of single variable calculus (differentiation and integration) are a prerequisite. Resource Guide to Vector Calculus. They consist largely of the material presented during the lectures Course Description. pdf from MAT 1236 at Edith Cowan University. 65 Lecture 11. Published: March 01, 2020. You Differential Calculus Lecture Notes Veselin Jungic & Jamie Mulholland Department of Mathematics Simon Fraser University c Jungic/Mulholland, August MAT203 Advanced Vector Calculus. I believe an interested student can Vector calculus is one of the most useful branches of mathematics for game development. Class : Lecture Scheme 2: 5% participation, 25% Homework, 30% Best Midterm, 40% Final Exam. Topics covered in these notes include the un-typed lambda calculus Vector Calculus 1 multivariable calculus 1. M, 2:50pm-4:30pm. Dynamical systems, Spring 2005 (183 pages) Linear Algebra (21b, Spring 2018) College Multivariable, (Fall 2017) Calculus Defines vectors, vector addition and vector subtraction. Lecture Vector Calculus - Fall 2011 Meetings. Tensor Calculus (The Dual of a Vector Space) Tensor Calculus 4-5 (Tensors as Multilinear Maps; Integral Curves; The Commutator) Tensor Calculus Vector Calculus (Multivariate Calculus)B SC 4th Semester (CBCS)Mathematics (Honours)MAT-HC-4016Lecture 1 Vectors in Euclidean Space 1. Lecture 7: Parametric Equations And Vectors: Example 2. g. To enlighten the learners in the concept of differential equations and multivariable calculus Scheme 2: 5% participation, 25% Homework, 30% Best Midterm, 40% Final Exam. coursera. The main concepts that will be covered are: • Coordinate transformations • Matrix operations • Scalars and vectors • Vector calculus Multivariable Calculus Lectures Richard J. Vector Space. But like wolfsy said, if you are trying to Lecture Notes on Variational and Tensor Calculus. A two-dimensional vector ﬁeld is a function f that maps each point (x,y) in R2 to a two-dimensional vector hu,vi, and similarly a three-dimensional vector ﬁeld maps (x,y,z) to hu,v,wi. You must substitute the parametric equations into both the vector field and position vector and then integrate. The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus MATH 252-3: Vector Calculus Course Webpage Quick Links: Library Reserves WebCT Course Webpage This Week Documents & Homework Information Vector Calculus (MAST20009) COVID-19 vaccination (or valid exemption) is a requirement for anyone attending our campuses. For Vector Calculus I like J. Vector Calculus previous lecture notes by Ben Allanach and Jonathan Evans ; Vector Calculus yet earlier lecture notes by Stephen Cowley. These theorems are needed in core engineering subjects such as Electromagnetism and Fluid Mechanics. Slide 19. A familiarity with some basic facts 1 17. Example : A~(x;y;z) = (x;xy;xz) (’(x;y;z) = x2yz) is a vector linear transformations and vector spaces are necessary ingredients for a proper discussion of ad-vanced calculus. Topics covered are Three Dimensional Space, Equations of normal vectors and tangent planes 24 1. Vector Fields 65 Vector Fields. Contents 1 Syllabus and Scheduleix Our last month will be combining the multivariate calculus with vector calculus and this culminates in several important theorems which tie all of Calculus Basic Concepts – In this section we will introduce some common notation for vectors as well as some of the basic concepts about vectors such as the magnitude of a vector and unit vectors. Home New to This Edition WebAssign for Vector Calculus. V. Hinglish Physics. Lecture 6: Addition Of Vectors MATH 221 { 1st SEMESTER CALCULUS LECTURE NOTES VERSION 2. Unit vectors, vectors Lecture 1 Vectors View this lecture on YouTube We deﬁne a vector in three-dimensional Euclidean space as having a length (or magnitude) and a direction. Knill you can ﬁnd plenty of exercises, lecture edge of vector calculus and real analysis, some basic elements of point set topology and linear algebra. Tromba (UCSC). In organizing this lecture note, I am indebted by Cedar Crest College Calculus IV Lecture Course Description. C Carter Lecture 11 where i j is the angle between two vectors iand j, and ij k is the angle between the vector kand the plane spanned by iand j, is equal to the parallelepiped that has ~a, ~b, and ~cemanating from its bottom-back corner. Course objective : To apply the basic concepts found in a first year calculus course to multivariable functions (limits, differentiation, and integration). edu Office Hours: Room 2030 – A free PowerPoint 3–1 Vector integrals; the line integral of ∇ψψ.

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# Stochastic matrix

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2010 Mathematics Subject Classification: Primary: 15B51 Secondary: 60J10 [MSN][ZBL]

A stochastic matrix is a square (possibly infinite) matrix $P=[p_{ij}]$ with non-negative elements, for which $$\sum_j p_{ij} = 1 \quad \text{for all i.}$$ The set of all stochastic matrices of order $n$ is the convex hull of the set of $n^n$ stochastic matrices consisting of zeros and ones. Any stochastic matrix $P$ can be considered as the matrix of transition probabilities of a discrete Markov chain $\xi^P(t)$.

The absolute values of the eigenvalues of stochastic matrices do not exceed 1; 1 is an eigenvalue of any stochastic matrix. If a stochastic matrix $P$ is indecomposable (the Markov chain $\xi^P(t)$ has one class of positive states), then 1 is a simple eigenvalue of $P$ (i.e. it has multiplicity 1); in general, the multiplicity of the eigenvalue 1 coincides with the number of classes of positive states of the Markov chain $\xi^P(t)$. If a stochastic matrix is indecomposable and if the class of positive states of the Markov chain has period $d$, then the set of all eigenvalues of $P$, as a set of points in the complex plane, is mapped onto itself by rotation through an angle $2\pi/d$. When $d=1$, the stochastic matrix $P$ and the Markov chain $\xi^P(t)$ are called aperiodic.

The left eigenvectors $\pi = \{\pi_j\}$ of $P$ of finite order, corresponding to the eigenvalue 1: $$\label{eq1} \pi_j = \sum_i \pi_i p_{ij} \quad \text{for all}\ j\,,$$ and satisfying the conditions $\pi_j \geq 0$, $\sum_j\pi_j = 1$, define the stationary distributions of the Markov chain $\xi^P(t)$; in the case of an indecomposable matrix $P$, the stationary distribution is unique.

If $P$ is an indecomposable aperiodic stochastic matrix of finite order, then the following limit exists: $$\label{eq2} \lim_{n\rightarrow\infty} P^n = \Pi,$$ where $\Pi$ is the matrix all rows of which coincide with the vector $\pi$ (see also Markov chain, ergodic; for infinite stochastic matrices $P$, the system of equations \ref{eq1} may have no non-zero non-negative solutions that satisfy the condition $\sum_j \pi_j < \infty$; in this case $\Pi$ is the zero matrix). The rate of convergence in \ref{eq2} can be estimated by a geometric progression with any exponent $\rho$ that has absolute value greater than the absolute values of all the eigenvalues of $P$ other than 1.

If $P = [p_{ij}]$ is a stochastic matrix of order $n$, then any of its eigenvalues $\lambda$ satisfies the inequality (see [MM]): $$\left| \lambda - \omega \right| \leq 1-\omega, \quad \text{where \omega = \min_{1 \leq i \leq n} p_{ii}.}$$ The union $M_n$ of the sets of eigenvalues of all stochastic matrices of order $n$ has been described (see [Ka]).

A stochastic matrix $P=[p_{ij}]$ that satisfies the extra condition $$\sum_i p_{ij} = 1 \quad \text{for all j}$$ is called a doubly-stochastic matrix. The set of doubly-stochastic matrices of order $n$ is the convex hull of the set of $n!$ permutation matrices of order $n$ (i.e. doubly-stochastic matrices consisting of zeros and ones). A finite Markov chain $\xi^P(t)$ with a doubly-stochastic matrix $P$ has the uniform stationary distribution.

#### References

 [G] F.R. Gantmacher, "The theory of matrices" , 1 , Chelsea, reprint (1977) (Translated from Russian) MR1657129 MR0107649 MR0107648 Zbl 0927.15002 Zbl 0927.15001 Zbl 0085.01001 [B] R. Bellman, "Introduction to matrix analysis" , McGraw-Hill (1960) MR0122820 Zbl 0124.01001 [MM] M. Marcus, H. Minc, "A survey of matrix theory and matrix inequalities" , Allyn & Bacon (1964) MR0162808 Zbl 0126.02404 [Ka] F.I. Karpelevich, "On the characteristic roots of matrices with non-negative entries" Izv. Akad. Nauk SSSR Ser. Mat. , 15 (1951) pp. 361–383 (In Russian)

Given a real $n\times n$ matrix $A$ with non-negative entries, the question arises whether there are invertible positive diagonal matrices $D_1$ and $D_2$ such that $D_1AD_2$ is a doubly-stochastic matrix, and to what extent the $D_1$ and $D_2$ are unique. Such theorems are known as $DAD$-theorems. They are of interest in telecommunications and statistics, [C], [F], [Kr].
A matrix $A$ is fully decomposable if there do not exist permutation matrices $P$ and $Q$ such that $$PAQ = \left[ \begin{array}{cc} A_1 & 0 \\ B & A_2 \end{array} \right].$$ A $1 \times 1$ matrix is fully indecomposable if it is non-zero.
Then for a non-negative square matrix $A$ there exist positive diagonal matrices $D_1$ and $D_2$ such that $D_1AD_2$ is doubly stochastic if and only if there exist permutation matrices $P$ and $Q$ such that $PAQ$ is a direct sum of fully indecomposable matrices [SK], [BPS].


# Sum of the inverse of a geometric series?

I'm trying to solve for this summation:

$$\sum_{j=0}^{i} {\left(\frac 1 2\right)^j}$$

This looks a lot like a geometric series, but it appears to be inverted. Upon plugging the sum into Wolfram Alpha, I find the answer to be

$$2-2^{-i}$$

but I don't understand how it gets there. Am I able to consider this a geometric series at all? It almost seems closer to the harmonic series.

• Welcome to Math Stack Exchange. Note $\frac 1 {2^j}=\left(\frac1 2\right)^j$ – J. W. Tanner Mar 25 '19 at 3:36
• Is $i$ some finite number? Or are you trying to find the value of the series for any arbitrary $i$? – kkc Mar 25 '19 at 3:36
• The reciprocals of each term of a geometric series is also a geometric one – lab bhattacharjee Mar 25 '19 at 3:37
• i is finite but arbitrary. The sum does not approach infinity. – bpryan Mar 25 '19 at 3:39
• My point of noting $\frac 1 {2^j}=\left(\frac 1 2\right)^j$ was not that $\frac 1 {2^j}$ was incorrect but rather that this is a geometric series – J. W. Tanner Mar 25 '19 at 3:49

$$\sum_{j=0}^{i} \frac{1}{2^j}=\sum_{j=0}^{i}\left( \frac{1}{2}\right)^j=\frac{1-\left(\frac12\right)^{i+1}}{1-\frac12}=2\left(1-\left(\frac12\right)^{i+1}\right)=2-\left(\frac12\right)^{i}=2-2^{-i}$$
As $$\frac1{2^j}=\left(\frac12\right)^j$$, this is also a geometric sum with the common ratio of $$r=\frac12$$. So you apply the formula for geometric sums $$\sum_{j=0}^nr^j=\frac{1-r^{n+1}}{1-r}$$to obtain the answer you have written.


# Does the category of local rings with residue field $F$ have an initial object?

Let $$F$$ be a field. Does the category $$C_F$$ of local rings with residue field isomorphic to $$F$$ have an initial object?

This is, for instance, true if $$F=\mathbb{F}_{p}$$ for some prime $$p$$: If $$R$$ is a local ring with residue field $$\mathbb{F}_{p}$$, then any $$x\in\mathbb{Z}\setminus(p)$$ must map to something invertible under the morphism $$\mathbb{Z}\longrightarrow R$$. Hence that morphism factors as $$\mathbb{Z}\longrightarrow\mathbb{Z}_{(p)}\longrightarrow R$$; thus $$\mathbb{Z}_{(p)}$$ is the initial object.

But what happens in the more general case? I guess it should be true at least if $$F$$ is of finite type over $$\mathbb{Z}$$, but I have no idea how to prove it.

(EDIT - To avoid any confusion: I am talking about an initial object in the category of local rings $$R$$ with a fixed surjection $$R\longrightarrow F$$.)

• That's a good question. It holds for $F=\mathbb Q$ as well – Maxime Ramzi May 15 '20 at 7:00
• @GeorgesElencwajg : I guess it depends on how you set up the category $C_F$ exactly : if it's just a full subcategory of rings, then you are right (composing $R\to F\to F$ gives two different morphisms $R\to F$ because $R\to F$ is surjective); but if you set it up as a subcategory of $CRing/F$ instead, it's not that clear. – Maxime Ramzi May 15 '20 at 7:27
• @Georges : I'm not saying $F$-algebras, on the other hand "the category of local fields with residue field $F$" could definitely mean "local rings $R$ with a fixed map $R\to F$ which exhibits $F$ as the residue field". So a subcategory of $CRing/F$, not $F/Cring$ – Maxime Ramzi May 15 '20 at 8:00
• @Georges: I meant it the way Maxime says it. – The Thin Whistler May 15 '20 at 8:10

Let $$\mathbb{F_4}=\{0,1,w,1+w\}$$ be the field of 4 elements. Suppose $$R$$ is the initial object in the category described in the question for the field $$\mathbb{F_4}$$. Then $$R$$ must contain some element $$x$$ which maps to $$w\in\mathbb{F_4}$$. Thus we have a map $$f\colon S\to R$$, where $$S=\mathbb{Z}[y]_M$$, sending $$y \mapsto x$$. Here $$M$$ is the maximal ideal of $$\mathbb{Z}[y]$$ containing $$2,1+y+y^2$$.

The following composition must be the identity: $$R \to S \stackrel f \to R$$ Thus $$R=S/I$$ for some ideal $$I\subset M$$. Further we know $$I\neq 0$$ as $$S$$ cannot be the initial object: there are multiple distinct maps $$S\to S$$, such as the identity map and the map sending $$y\mapsto y+2$$.

Under the composition $$S \stackrel f \to R\to S$$, we have $$y\mapsto p/q$$, for some $$p,q$$ integer polynomials in $$y$$. We know $$p/q$$ is not a rational number as $$p/q\mapsto w\in\mathbb{F_4}$$. Thus $$p/q$$ is a non-constant rational function in one variable, taking infinitely many values, which cannot all satisfy the same polynomial over the integers.

On the other hand, as $$I\neq 0$$ there must be a polynomial over the integers satisfied by $$p/q$$. This gives us the desired contradiction.

• This is so clever! How did you come up with this approach? – diracdeltafunk May 16 '20 at 2:26
• Thankyou. It felt like if there was an initial object it ought to be S, but at the same time it couldn't be. Playing those off against each other and using that S is a well behaved, concretely described ring led to the contradiction. – tkf May 16 '20 at 3:55

The category $$C_{F}$$ possesses a weak initial object $$I_{F}$$, i.e. an object that is unique up to not necessarily unique isomorphism.

Let $$F$$ be a field and $$L$$ be its minimal subfield (the smallest subfield contained in $$F$$). Then either $$L=\mathbb{F}_{p}$$ for some prime $$p$$ or $$L=\mathbb{Q}$$.

Assume first that $$F$$ is of finite type over $$L$$. Let $$n\in\mathbb{N}$$ be the smallest natural number so that $$F=L[x_{1},...,x_{n}]/\mathfrak{m}$$ for some maximal ideal $$\mathfrak{m}\subseteq L[x_{1},...,x_{n}]$$. Let $$\overline{x}_{i}$$ be the image of $$x_{i}\in L[x_{1},...,x_{n}]$$ in $$F$$.

Let $$\zeta:R\longrightarrow F$$ be a surjection where $$R$$ is a local ring. Since every $$\overline{x}_{i}$$ has a (not necessarily unique) preimage $$\zeta^{-1}(\overline{x}_{i})\in R$$, there is a (not necessarily unique) morphism $$\kappa:\mathbb{Z}[x_{1},...,x_{n}]\longrightarrow R$$ that fits into a commutative diagram $$\require{AMScd}$$ $$\begin{CD} \mathbb{Z}[x_{1},...,x_{n}]@>{\kappa}>>R\\ @V{\pi}VV @VV{\zeta}V\\ L[x_{1},...,x_{n}] @>>{\chi}>F \end{CD}$$ Let $$\mathfrak{i}:=\chi^{-1}\pi^{-1}(0)=\pi^{-1}(\mathfrak{m})$$. The ideal $$\mathfrak{i}$$ is always prime; it is maximal if and only if $$L=\mathbb{F}_{p}$$ for some prime $$p$$. Since $$R$$ is local, every element of $$\mathbb{Z}[x_{1},...,x_{n}]$$ is mapped by $$\kappa$$ onto something invertible in $$R$$. Hence $$\kappa$$ factors as $$\begin{CD} \mathbb{Z}[x_{1},...,x_{n}] @>>>\mathbb{Z}[x_{1},...,x_{n}]_{(\mathfrak{i})} @>{\lambda}>> R \end{CD}$$ Thus $$I_{F}:=\mathbb{Z}[x_{1},...,x_{n}]_{(\mathfrak{i})}$$ is a weak initial object in the category $$C_{F}$$.

Note that the assignment $$\kappa\longleftrightarrow\lambda$$ is unique in both ways: To each choice of $$\kappa$$ there is a unique $$\lambda$$ and vice-versa.

Assume next that $$F$$ is of infinite type over $$L$$. Then $$F$$ is the direct limit of all morphisms $$F'\longrightarrow F''$$, where $$F',F''$$ are fields of finite type over $$L$$. Since the construction of $$I_{-}$$ is functorial and compatible with direct limits, $$I_{F}$$ can be defined as $$I_{F}:=\lim_{F'\text{ of fin. t.}/L}I_{F'}$$.

The initial object is strong, i.e. unique up to unique isomorphism, if and only if $$F=L$$.

Namely, if $$F=L$$, then $$n=0$$ and the unique morphism $$\kappa:\mathbb{Z}\longrightarrow R$$ induces a unique morphism $$\lambda:\mathbb{Z}_{(\mathfrak{i})}\longrightarrow R$$.

Else, if $$F\neq L$$, then $$n\geq 1$$ and for any $$i\in\{1,...,n\}$$ and any $$s\in\mathfrak{i}\setminus\{0\}$$, the map $$\xi_{i,s}:x_{i}\mapsto x_{i}+s$$ yields a nontrivial automorphism $$I_{F}\longrightarrow I_{F}$$ that commutes with the surjection $$I_{F}\longrightarrow F$$.

My guess is that the $$\xi_{i,s}$$ actually generate the whole group $$\operatorname{Aut}(I_{F})$$, but I have yet to figure out a proof for this...


# Find an equivalent sequence as $n\to +\infty$ of $u_1>0, u_{n+1} = \frac{u_n}{n} + \frac{1}{n^2}$

Let $$u_1>0$$ be a real number. Let us consider $$(u_n)_{n\geq 1}$$ the sequence such as:

$$\forall n \geq 1, u_{n+1} = \frac{u_n}{n} + \frac{1}{n^2}\quad (\star)$$

Find an equivalent of $$u_n$$ as $$n\to +\infty$$.

So I found a way to show that $$u_n \sim \frac{1}{n^2}$$, but I'm quite unhappy with this method because I feel like I found it by chance without understanding anything (I did a lot of trials and found this)

My method:

I showed by induction that $$u_n \leq (u_1+1)$$. Thus, $$u_n\to 0$$ considering $$(\star)$$.

Then, $$nu_{n+1} = u_n + 1/n$$. Thus (since $$n+1 \sim n$$), $$nu_n \to 0$$.

To end with, I have $$n^2u_{n+1} = nu_n + 1$$. Thus, $$(n+1)^2 u_n \sim n^2u_n \to 1$$.

How would you solve such a problem? Is there any more intuitive method that one may have done?

• It is right actually. I think elementary proof has you find is rather good. Where the intuition comes it that $u_n$ is ponderated with $n$ and $u_{n+1}$ can be with $n^2$ by multiplying all the equation by $n^2$. So I think that ponderation lead you to the equivalent. – EDX Jun 4 '20 at 15:39

1. There is a heuristic argument which is useful for guessing the behavior of $$u_n$$: Rewrite the recurrence relation as

$$u_{n+1} - u_n = \frac{u_n}{n} - u_n + \frac{1}{n^2}.$$

Its continuum analogue is the following differential equation:

$$y' = \frac{y}{x} - y + \frac{1}{x^2}.$$

Using the standard method, this equation can be solved as:

$$y(x) = x e^{-x} \int \frac{e^x}{x^3} \, \mathrm{d}x.$$

Then L'Hospital's Rule then tells that $$y(x) \sim x^{-2}$$ as $$x \to \infty$$. From this observation, we may as well expect that $$u_n \sim n^{-2}$$.

2. The above ansatz suggests that, in the recurrence relation for $$u_{n+1}$$, $$\frac{1}{n^2}$$ is the dominating term and $$\frac{u_n}{n}$$ is much smaller as $$n\to\infty$$. In particular, nesting this relation will produce an expansion with ever decreasing terms. This idea can be easily tested as follows:

Let $$r_n = (n-1)u_n$$. Then for $$n \geq 2$$,

$$r_n = (n-1)u_{n} = \frac{1}{n-1} + \frac{r_{n-1}}{n-1}.$$

From this, we get

\begin{align*} r_n &= \frac{1}{n-1} + \frac{r_{n-1}}{n-1} \\ &= \frac{1}{n-1} + \frac{1}{(n-1)(n-2)} + \frac{r_{n-2}}{(n-1)(n-2)} \\ &= \frac{1}{n-1} + \frac{1}{(n-1)(n-2)} + \frac{1}{(n-1)(n-2)(n-3)} + \frac{r_{n-3}}{(n-1)(n-2)(n-3)} \\ &\qquad\vdots\\ &= \sum_{k=1}^{n-2} \frac{1}{(n-1)\cdots(n-k)} + \frac{r_2}{(n-1)!} \end{align*}

Using this, it is not hard to conclude that $$(n-1)^2 u_n \to 1$$ as $$n\to\infty$$, and in fact, we can extract an asymptotic expansion of $$u_n$$ up to any prescribed order $$\mathcal{O}(n^{-M})$$.

This is similar to Sangchul Lee's approach.

If we multiply $$n!$$ both sides of the recurrence, we obtain $$n!u_{n+1}=(n-1)!u_n+\frac{n!}{n^2}.$$ Applying the above repeatedly, $$u_{n+1}=\frac1{n!} \sum_{k=1}^n \frac{k!}{k^2} + \frac{u_1}{n!}.$$

• That's a really good method! Thanks – MiKiDe Jun 7 '20 at 9:17


# Given $3$ types of coins, how many ways can one select $20$ coins so that no coin is selected more than $8$ times.

So I was given this question.

Given $3$ types of coins, how many ways can one select $20$ coins so that no coin is selected more than $8$ times.

First I make $x_1 + x_2 + x_3 = 20$ Then $0 \leq x_i \leq 8$

Then we use the Inclusion exclusion principle

Let $A_i$ be the non-negative integer solutions to $x_1 + x_2 + x_3 = 20$ $x_i \geq 9$. Then use inclusion exclusion formula to find $N(A_1 \bigcap A_2 \bigcap A_3)$

What i don't get is how to apply the inclusion exclusion formula. I know the process to get to it but not how to apply it.

• Possible duplicate of Inclusion-exclusion principle: Number of integer solutions to equations – Cloverr Feb 2 '16 at 18:27
• You can also find the coefficient of $x^{20}$ in $(1+x+\ldots + x^8)^3$. Then you get the right answer as well. – Henno Brandsma Feb 2 '16 at 18:27
• @Nilanjan thats a different question – Zero Feb 2 '16 at 18:31
• Do you have to use inclusion-exclusion? The problem is small enough in scale to permit more straightforward approaches. Heck, even plain enumeration will do. – Brian Tung Feb 2 '16 at 18:56
• I've added inclusion-exclusion to my answer. Interestingly, the approach by generating function arrives at the same expression! – Brian Tung Feb 2 '16 at 19:03

Instead of trying to apply formulas take a look at the general theory. Suppose $a_j\le x_j\le b_j$ for $j=1,2,3$.

$$(x^{a_1}+x^{a_1+1}+\cdots+x^{b_1})(x^{a_2}+x^{a_2+1}+\cdots+x^{b_2})(x^{a_3}+x^{a_3+1}+\cdots+x^{b_3})$$

Every solution of $x_1+x_2+x_3=n$ with the constraints $a_j\le x_j\le b_j$ contributes $1$ to the coefficient of $x^n$ in the above expression. So the number of solutions is the coefficient of $x^n$.

Substituting accordingly we see we require the coefficient of $x^{20}$ in $$(1+x+\cdots+x^8)^3={(1-x^9)^3\over (1-x)^3}$$ or equivalently in $$(1-3x^9+3x^{18})\sum_{k=0}^{20}\binom{2+k}{k}x^k$$ which equals $$\binom{22}{20}-3\binom{13}{11}+3\binom{4}{2}$$

• The equation $x_1 + x_2 + x_3 = 20$ has only $\binom{22}{20}$ solutions in the non-negative integers. Check your calculations. – N. F. Taussig Feb 2 '16 at 18:42
• I replaced $3$ with $20$ erroneously.. – Jack's wasted life Feb 2 '16 at 18:45

There are $15$ different ways to select $20$ coins of three different types, with no type having more than $8$ specimens.

There are a number of different methods to obtain this number. One is to condition on the number of coins of Type $1$. If there are $4$ coins of Type $1$, then there is only one way to obtain $16$ coins of the other two types; if there are $5$ coins of Type $1$, then there are two ways to obtain $15$ coins of the other two types; and so on. $1+2+3+4+5 = 15$.

Another way is to consider that the solutions are the intersection of the lattice cube $\{(x, y, z)) \in \mathbb{N}_{\geq 0}^3 \mid 0 \leq x, y, z \leq 8\}$ with the plane $x+y+z = 20$. If you have any facility with this, you will notice that the intersection is a triangular lattice of side $5$, so the number of solutions is the $5$th triangular number, $15$.

ETA: Finally, we can use inclusion-exclusion, although I find it less straightforward. The number of non-negative integer solutions to $x+y+z = 20$ is, by the usual stars-and-bars approach, $\binom{22}{2} = 231$. However, this counts solutions where at least one of $x, y, z > 8$. The number of solutions where $x > 8$ is, again by stars-and-bars, $\binom{13}{2} = 78$, and there are three coin types, so we must subtract $3 \times 78 = 234$ to obtain $-3$.

Again, however, this double-counts three cases where at least two of $x, y, z > 8$. There are $\binom{4}{2} = 6$ ways in which both $x, y > 8$, but there are three different pairs of coin types, so we must add back $3 \times 6 = 18$ to get $15$. There are no ways to have $x, y, z > 8$ and still add up to $20$, so $15$ is the final result. Note, interestingly, that this is the same expression that Jack's wasted life arrived at, although that method was by way of generating functions.

• One typographical error: In the last sentence of the first paragraph of the inclusion-exclusion argument, you meant $x > 8$ rather than $x > 0$. – N. F. Taussig Feb 2 '16 at 20:14
• @N.F.Taussig: Thanks for that. – Brian Tung Feb 3 '16 at 18:58

Let $8-x_i=:y_i$ $(1\leq i\leq3)$. Then we want all $y_i\geq0$ and $y_1+y_2+y_3=4$. There are $${4+3-1\choose 3-1}={6\choose 2}=15$$ ways to choose admissible $y_i$.


IntMath Home » Forum home » Integration » Different parabola equation when finding area

# Different parabola equation when finding area [Solved!]

### My question

In chapter 3, compare example 1 to example 3. Why isnt (3 - x^2) the correct parabolic equation for example 3?

### Relevant page

3. Areas Under Curves

### What I've done so far

When I went to solve example 3, I used the equation (3 - x^2) for the parabola. I found out that was not the correct parabolic equation upon looking at your solution.
I am not sure why my equation cannot be used even though the vertex is not at (0,3), i.e. not lined up with the origin? The height is still three so I thought it would be okay.

X

In chapter 3, compare example 1 to example 3.  Why isnt (3 - x^2) the correct parabolic equation for example 3?
Relevant page

3. Areas Under Curves

What I've done so far

When I went to solve example 3, I used the equation (3 - x^2) for the parabola. I found out that was not the correct parabolic equation upon looking at your solution.
I am not sure why my equation cannot be used even though the vertex is not at (0,3), i.e. not lined up with the origin?  The height is still three so I thought it would be okay.

## Re: Different parabola equation when finding area

@phinah: Your parabola will certainly have the correct height, but let's consider its width. Where does your parabola intersect with the x-axis?

What width at the x-axis does this give you?

X

@phinah: Your parabola will certainly have the correct height, but let's consider its width. Where does your parabola intersect with the x-axis?

What width at the x-axis does this give you?

## Re: Different parabola equation when finding area

I realized after I inquired to you that this upside down parabola is a shift one unit to the right on the x-axis.

So I came up with the -(x-1)^2 + 3.

But the parabola bottom would start at (0,2) Would that be an issue?

NOTE: The integral would be (-x^2 + 2x + 2) dx for a value of 16/3 square units.

X

I realized after I inquired to you that this upside down parabola is a shift one unit to the right on the x-axis.

So I came up with the -(x-1)^2 + 3.

But the parabola bottom would start at (0,2) Would that be an issue?

NOTE: The integral would be (-x^2 + 2x + 2) dx for a value of 16/3 square units.

## Re: Different parabola equation when finding area

@phinah: I edited your answer to include math formatting, which you are encouraged to do.

Once again, the width at the x-axis is a problem (both with your original attempt and with this revision.)

a. Where does your parabola intersect with the x-axis when we use y=3-x^2?

b. What width at the x-axis does this give you?

You may find some of the background in this article useful: How to find the equation of a quadratic function from its graph

X

@phinah: I edited your answer to include math formatting, which you are encouraged to do.

Once again, the width at the x-axis is a problem (both with your original attempt and with this revision.)

a. Where does your parabola intersect with the x-axis when we use y=3-x^2?

b. What width at the x-axis does this give you?

You may find some of the background in this article useful: How to find the equation of a quadratic function from its graph

## Re: Different parabola equation when finding area

3 - x^2 = 0
 x = +/- sqrt 3'
So my length is 3.46 when it should only be 2.

X

3 - x^2 = 0
 x = +/- sqrt 3'
So my length is 3.46 when it should only be 2.

## Re: Different parabola equation when finding area

Correction

x = + or - sqrt 3

X

Correction

x = + or - sqrt 3

## Re: Different parabola equation when finding area

@Phinah: Yes, correct, so it's not possible to use y=3-x^2 for your function. But it is possible to use one that passes through (0, 3). Can you find it?

BTW, to achieve "plus or minus" in the math entry system, you enter

+-

So your answer would look like: x = +- sqrt(3).

You can see all the syntax for mathematical symbols here: ASCIIMath syntax.

X

@Phinah: Yes, correct, so it's not possible to use y=3-x^2 for your function. But it is possible to use one that passes through (0, 3). Can you find it?

BTW, to achieve "plus or minus" in the math entry system, you enter

+-

So your answer would look like: x = +- sqrt(3).

You can see all the syntax for mathematical symbols here: ASCIIMath syntax.

## Re: Different parabola equation when finding area

I cannot.

X

I cannot.

## Re: Different parabola equation when finding area

Well, since we know the width of the arch must be 2 m, then we know it must go through (-1, 0) and (1, 0).

So using the general form of a quadratic,

y = ax^2 + bx + c,

you can substitute the 3 known points in and thus find the values of a, b and c.

Have a go!

X

Well, since we know the width of the arch must be 2 m, then we know it must go through (-1, 0) and (1, 0).

So using the general form of a quadratic,

y = ax^2 + bx + c,

you can substitute the 3 known points in and thus find the values of a, b and c.

Have a go!

## Re: Different parabola equation when finding area

@murray: phinah hasn't responded. Would you like me to answer this?

X

@murray: phinah hasn't responded. Would you like me to answer this?

## Re: Different parabola equation when finding area

X

@stephenB: Sure, please go ahead!

## Re: Different parabola equation when finding area

Okay, so we have the general formula

y = ax^2 + bx + c,

and the 3 points it needs to go through, (-1, 0), (1, 0) and (0, 3).

Plugging them in:

0 = a(1)^2 + b(1) + c  = a + b + c ... [1]

0 = a(-1)^2 + b(-1) + c  = a - b + c ... [2]

3 = a(0)^2 + b(0) + c = c ... [3]

0 = 2a + 2c which means a + c = 0 ...[4]

From [3] we know c=3, so from [4], a=-3 and then plugging those into [1] gives b=0.

That is, the parabola's equation is:

y = -3x^2 + 3

X

Okay, so we have the general formula

y = ax^2 + bx + c,

and the 3 points it needs to go through,  (-1, 0),  (1, 0) and (0, 3).

Plugging them in:

0 = a(1)^2 + b(1) + c  = a + b + c ... [1]

0 = a(-1)^2 + b(-1) + c  = a - b + c ... [2]

3 = a(0)^2 + b(0) + c = c ... [3]

0 = 2a + 2c which means a + c = 0 ...[4]

From [3] we know c=3, so from [4], a=-3 and then plugging those into [1] gives b=0.

That is, the parabola's equation is:

y = -3x^2 + 3

## Re: Different parabola equation when finding area

Thanks, stephenB.

I think if phinah tries the equation you found, he will find the area of the parabolic glass pane will the the same as in my solution.

X

Thanks, stephenB.

I think if phinah tries the equation you found, he will find the area of the parabolic glass pane will the the same as in my solution.

## Re: Different parabola equation when finding area

Phinah DID respond with the correct solution Murray BEFORE Stephen did. I do not know why it did not come through

X

Phinah DID respond with the correct solution Murray BEFORE Stephen did.  I do not know why it did not come through

## Re: Different parabola equation when finding area

Hi Phinah. Sorry - it must have been a temporary database glitch or something.

X

Hi Phinah. Sorry - it must have been a temporary database glitch or something.

## Re: Different parabola equation when finding area

Okay Murray. Thank you.

X

Okay Murray.  Thank you.


# Partial sum of the harmonic series between two consecutive fibonacci numbers

I was playing around with some calculations and I noticed that the partial sum of the harmonic series: $$s_n=\sum_{k=F_n}^{F_{n+1}}\frac 1 k$$ where $F_n$ and $F_{n+1}$ are two consecutive Fibonacci numbers have some interesting properties. It is close to $\frac 1 2$ for small values of $n$ and it seems to converge to a value less than $0.5$ for large $n$. This is what I've got so far: $$\lim_{n\to\infty} s_n\approx 0.481212$$ I googled a bit to see if there is some theorems or resources for this, and found nothing. I suspect that the series might converge to a smaller number and I may have reached some computational limitations which led to the conclusion that the limit is close to $\frac 1 2$. So my questions are:

1. Can we show that the series converge to a non-zero value?
2. In case the first answer is yes, can the limit be expressed in a closed form?

In terms of the harmonic numbers $H_n$, your sequence is

$$s_n = H_{F_{n+1}} - H_{F_n-1}$$

As $n \to \infty$ it's known that $H_n = \log n + \gamma + o(1)$, so

\begin{align} s_n &= \log F_{n+1} + \gamma + o(1) - \log(F_n-1) - \gamma - o(1) \\ &= \log F_{n+1} - \log(F_n-1) + o(1). \end{align}

Now $F_m \sim \varphi^m/\sqrt{5}$, where $\varphi$ is the golden ratio, so using the fact that $a \sim b \implies \log a = \log b + o(1)$ we have

\begin{align} s_n &= \log(\varphi^{n+1}/\sqrt{5}) - \log(\varphi^{n}/\sqrt{5}) + o(1) \\ &= \log \varphi + o(1). \end{align}

In other words,

$$\lim_{n \to \infty} \sum_{k=F_n}^{F_{n+1}} \frac{1}{k} = \log \varphi.$$

• It might be better to use $s_n\approx ...$ instead of $s_n=...$ – polfosol Oct 30 '16 at 8:14
• @polfosol, no I disagree. Everything in my answer is rigorous following the definitions of $\sim$ and little-o notation. – Antonio Vargas Oct 30 '16 at 8:14
• @polfosol, see, for example, here for the definitions. – Antonio Vargas Oct 30 '16 at 8:15
• I didn't notice that. Fair enough – polfosol Oct 30 '16 at 8:15
• I will add a comment that ought to have been done : thanks for your very precise and nice answer. – Jean Marie Oct 30 '16 at 8:35

The Fibonacci numbers increase as $\phi^n$ (where $\phi$ is the golden mean $\frac{1+\sqrt{5}}{2}$), and harmonic numbers increase as $\log n$ (i.e., the natural log). Therefore, the difference between the harmonic numbers for successive Fibonacci numbers will approach $\log\phi \approx 0.481211825...$

To expand a bit, the Fibonacci numbers can be expressed as $\frac{\phi^n - (1-\phi)^n}{\sqrt{5}}$. (Try it! The fact that the equation $f(x+2) - f(x+1) - f(x) = 0$ requires a sum of powers of $\phi$ and $1-\phi$ follows from the fact that these are the solutions to the equation $x^2 - x - 1 = 0$, and the coefficients come from f(1) = f(2) = 1.) The second term vanishes, so large Fibonacci numbers can be approximated quite well as $\frac{\phi^n}{\sqrt{5}}$.

Since one definition of the natural logarithm is the integral from 1 to the parameter of the function $t^{-1}$, the harmonic numbers can be approximated as the natural logarithm, and in fact the difference approaches a constant (called $\gamma$, about 0.577). If you're not familiar with integrals, the fact that the harmonic numbers increase as a logarithm is suggested by Oresme's proof that the harmonic series diverges...

$$1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} + \frac{1}{9} + \cdots > 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{4} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{8} + \frac{1}{16} + \cdots$$

...and it just so happens that that logarithm is the natural logarithm.

So if you accept that for very large n, the harmonic numbers approach $\log n$, and that the Fibonacci numbers approach $\frac{\phi^n}{\sqrt{5}}$, then you get for two successive...

$$\log\left(\frac{\phi^{n+1}}{\sqrt{5}}\right) - \log\left(\frac{\phi^n}{\sqrt{5}}\right) = \log\left(\frac{\phi^{n+1}}{\phi^n}\right) = \log\phi$$

($\log x - \log y = \log \frac{x}{y}$ is a natural inverse of $\frac{e^x}{e^y} = e^{x-y}$.)

• I will mark this as accepted if you add some more details ;) – polfosol Oct 30 '16 at 8:10
• ...har-r-r-rumph. – user361424 Oct 30 '16 at 8:36
• @user361424 very nice answer, a compliment that ought to have been done by the proposer before asking for "more details" ! – Jean Marie Oct 30 '16 at 8:42
• @JeanMarie It seems I am stuck with this – polfosol Oct 30 '16 at 8:44
• I think this might be the inverse fastest gun in the west problem... (this answer was originally just the first paragraph, and without the parentheticals explaining the golden mean and clarifying the natural log). – user361424 Oct 30 '16 at 8:45