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Vector Calculus Pdf

Although he did his work before the invention of calculus, we can more easily develop his theory, as Newton did, with multivariate calculus. A vector whose divergence is zero then it is said to be divergence free vector (or) solenoid vector i. 2Vector Calculus. 3 Vector Calculus In the last part of the course, we will study vector elds, which are functions. (vector calculus), vector_calculus calculus, vector calculus calculus — Translations: Векторен анализ, Vektoranalysis, Cálculo vectorial, Analyse vectorielle, Vektorska analiza, Calcolo vettoriale. If you have a copy of Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach, we invite you to write [email protected] Otherwise it is impossible to understand. Geared towards students taking courses. Vector Algebra A vector has direction and magnitude both but scalar has only magnitude. The chain rule can be extended to the vector case using Jacobian matrices. The del-operator may be applied. Most likely you have knowledge that, people have see numerous times for their favorite books subsequent to this Textbook of Vector Calculus, but. This text explains the theory of vector-valued functions, with detailed solutions of all It also reviews relevant topics in linear algebra and calculus of real-valued. It is essential that these are studied carefully, so that a full un derstanding is developed before moving ahead. The divergence of a vector is a scalar result, and the divergence of a 2nd order tensor is a vector. The graph of a function of two variables, say, z=f(x,y), lies in Euclidean space, which in the Cartesian coordinate system consists of all ordered triples of real numbers (a,b,c). READ PAPER. These notes provide a quick review and summary of the concepts of vector calculus as used in electromagnetism. pdf] - Read File Online - Report Abuse. In vector (or multivariable) calculus, we will deal with functions of two or three variables (usually x,y or x,y,z, respectively). Elementary Calculus Home Page; Trigonometry Home Page. Olver University of Minnesota 1. Vector Calculus: Marsden, University Jerrold E, Tromba, Anthony: 9781429215084: Books - Amazon. There are four examples sheets for this course. Most likely you have knowledge that, people have see numerous times for their favorite books subsequent to this Textbook of Vector Calculus, but. Did you like this article on. approaching. zip Kreyszig – Advanced Engineering Mathematics. Now suppose we have a scalar function f (r) and we. 1 Ellipsoids 12. Vector Calculus. 6 Reciprocal System of Vectors 1. Vectors in two and three dimensions, equations of lines and planes, functions of several variables, partial differentiation, directional derivatives, gradients, optimization with Lagrange multipliers, multiple integration, line integrals and vector fields. Professor. org September 26, 2018. edu) O ce J, Pink Side Thomas Shi ey (schi [email protected] The divergence of a vector is written as $$abla \cdot {\bf v}$$, or $$v_{i,i}$$ in tensor notation. Tutors offer focused, interactive learning environments where you can explore and reinforce fundamental concepts. dimensions (. org on March 20, 2021 by guest [MOBI] Textbook Of Vector Calculus Thank you totally much for downloading Textbook of Vector Calculus. Download Full PDF Package. Author: Dhanalakshmi Created Date: 20170422193809Z. 4MB, PDF) Download mirrors: Mirror 1. 6 Parametric Surfaces and Their Areas 17. Craig Beasley Department of Electrical and Systems Engineering Vector calculus is a staple of the engineering disciplines. Includes index. Vector calculus is the fundamental language of mathematical physics. 4 Green’s Theorem and applications. •Theorem: If F is vector field defined on whose component functions have continuous partial derivatives and , then F is a conservative vector field. pdf ORDINARY DIFFERENTIAL EQUATIONS. Margham Publications, Chennai. Many quantities which are of interest in physics are both directed quantities (vectors) and can take on a continuous range of values, making calculus methods necessary. The Þrst-order diﬀerential operator ∇ (called ﬁdelﬂ) has three components: ∇ = • ∂ ∂x, ∂ ∂y, ∂ ∂z ‚ where are unit vectors in the x,y,andz directions respectively. Mathematics Study Materials VECTOR CALCULUS. The book's careful account is a contemporary balance between theory, application, and historical development, providing its readers with an insight into how mathematics progresses and is in turn influenced by the natural world. MANMOHAN DASH, PHYSICIST, TEACHER ! Physics for ‘Engineers and Physicists’ “A concise course of important results” Lecture - 1 Vector Calculus and Operations Lectures around 9. A vector field in the plane can be visualised as a collection of arrows with a given magnitude and direction, each attached to a point in the plane. As understood, talent does not suggest that you have astonishing points. This bestselling vector calculus text helps students gain a solid, intuitive understanding of this important subject. Vectors and Surfaces. pdf from CS 289A at University of California, Berkeley. 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. Download full Text Book Of Vector Calculus books PDF, EPUB, Tuebl, Textbook, Mobi or read online Text Book Of Vector Calculus anytime and anywhere on any device. Click below to download the previous version of the Calculus Volume 3 PDF. Lectures 1 & 2 - Review of Vector Calculus This is a quick review of some of the major concepts in vector calculus that is used in this class. Sergei Suslov Office: PSA 621. magnet:?xt=urn:btih:dc5aaacfc2. Books selection & Best Mathematics Books play invaluable role to increase chances of success. This volume contains frequent references to Volume 1. In vector calculus, spatial derivatives are performed on vector AND scalar Þelds to derive other vector or scalar Þelds. Calculus and Vectors. Hubbard and B. The Divergence Theorem states that if is an oriented closed surface in 3 and is the region enclosed by and F is a vector ﬁeld whose components. Download Full PDF Package. We have Vector Calculus (2nd Edition) txt, PDF, doc, ePub, DjVu forms. edu office: APM 5256, Office hours: M:3:30-4:30, W:4:30-5:30 Please show up at least 15 minutes before the end of the office hour. Vector Calculus in Two Dimensions by Peter J. 2 Matrices and the Cross Product 12. Find an equation for the line through the point (3;¡1;2) and perpendic- ular to the plane 2x¡y +z +10 = 0. 2 Line Integrals 17. III Integration and Curvilinear Coordinates. Connections to the Study Design: AOS 4 – Vectors. Download or read it online for free here: Download link (2. Vec-tor/Matrix calculus extends calculus of one vari-able into that of a vector or a matrix of vari-ables. The cross product of two vectors + œ a+Bß +Cß +Db and , œ a,Bß ,Cß ,Db in Vector Calculus. MATH 2210-2220 uses tools and techniques developed in linear algebra (MATH 2210, taken first) to develop multivariable and vector calculus (MATH 2220). Includes index. Vector Calculus for Electromagnetism is the backbone of any comprehensive treatment of Electromagnetism. A short summary of this paper. dkarkada/coursenotes/Vector-Calculus. It has been used for the past few years here at Georgia Tech. The vector diﬀerential operator ∇, called “del” or “nabla”, is deﬁned in three dimensions to be: ∇ = ∂ ∂x i+ ∂ ∂y j + ∂ ∂z k. This version was used beginning in March 2018. Vector Calculus in Mathematica. [PDF]Calculus Volume 1 ( Solutions Manual ) by J. View vector-calculus-worksheet. 8 Stokes’ Theorem 17. 1 Line Integrals. @x is a M N matrix and x is an N-dimensional vector, so the product @y @x x is a matrix-vector multiplication resulting in an M-dimensional vector. 7THE DIVERGENCE OF A VECTOR FIELD When ris dotted with a vector ﬁeld a, it produces a scalar ﬁeld equal to the net ﬂux of a out of each point in space. In order to describe the motion of the air, the air velocity must be deﬁned as a function of position, which means that a velocity vector has to be assigned to every point in space. Find an equation for the line through the point (3;¡1;2) and perpendic- ular to the plane 2x¡y +z +10 = 0. Vector Calculus. Introduction;. Null Vector : A vector of zero magnitude, which can have no direction associated with it is called zero (or null) vector and is denoted by O-a thick zero. Marsden and Tromba Free shipping over 10. : In V3, 3 non-coplanar vectors are linearly independent; i. A vector field is uniquely specified by giving its divergence and curl within a region and its normal component over the. Now in its fifth edition, Vector Calculus helps students gain an intuitive and solid understanding of this important subject. Verify that this result holds when = x + y and V is the spherical volume x 2 +. CS 189 Introduction to Machine Learning Spring 2020 Jonathan Shewchuk 1 Vector Calculus Review Proving. Latest version (2021-01-02): VectorCalculus. single-variable calculus, and also relates to formulas for area and volume from MAT 169 that are de ned in terms of determinants, or equivalently, in terms of the dot product and cross product. The prerequisites are the standard courses in single-variable calculus (a. Download it Vector Calculus books also available in PDF, EPUB, and Mobi Format for read it on your Kindle device, PC, phones or tablets. aDepartment of Applied Mathematics and Theoretical Physics, Centre for Mathemati-cal Sciences, University of Cambridge, Wilberforce Road. However, beginners report various difficulties dealing with the index notation due to its bulkiness. Vector Calculus 2. As the set fe^ igforms a basis for R3, the vector A may be written as a linear combination of the e^ i: A= A 1e^ 1 + A 2e^ 2 + A 3e^ 3: (1. Here is a brief review,. This book introduces students to vector analysis, a concise way of presenting certain kinds of equations and a natural aid for forming mental pictures of physical. Vector Calculus Tutors for Students. 1b, or the left-handed screw, Fig A. 1 INTRODUCTION. ] INTRODUCTION Among the many alternative mathematical systems in which the equations of physics can be expressed, two of the most popular are the matrix form of spinor calculus and the vector calculus formulated by Gibbs. Vector Calculus in Mathematica. ) Vector Calculus,MarsdenandTromba Rigorous and enjoyable but slightly demanding. Vector Calculus 6th edition solutions are available for this textbook. Differential Forms A Complement to Vector Calculus, Steven H. The Divergence Theorem states that if is an oriented closed surface in 3 and is the region enclosed by and F is a vector ﬁeld whose components. Unit-VIII Vector Calculus. • When dealing with large and/or time varying objects or phenomenon we must be able to describe the state at locations, and as a whole. To maintain this website, we need your help. Vectors and Parametric Curves. Appendix D: MATRIX CALCULUS D–8 §D. Vector Calculus. Horizontal (i,j) parts rotating clockwise (compare with examples 3, 4 above). Leibnitz‟s theorem (without proof) – problems. School Sugar. The product of a vector A by a scalar m is a vector mA with magnitude Imf times the magni-tude of A and with direction the same as or opposite to that of A, according as m is positiveor. Click below to download the previous version of the Calculus Volume 3 PDF. 15 Scalar Equation of a Plane a(x x 0) + b(y y 0) + c(z z 0) = 0 where (x 0;y 0;z 0) is a point on the plane and ha;b;ciis the vector normal to the plane. pdf: File Size: 1441 kb: File Type: pdf: Download File. 15 Scalar Equation of a Plane a(x x 0) + b(y y 0) + c(z z 0) = 0 where (x 0;y 0;z 0) is a point on the plane and ha;b;ciis the vector normal to the plane. Mathematical Tripos: IA Vector Calculus Contents 0. This book presents the necessary linear algebra and then uses it as a framework upon which to build multivariable calculus. The print version of this textbook is ISBN: 9780321780652, 0321780655. This would be pictured by drawing the vector (1, 2, 1) in the opposite direction. Vector Calculus or Download Marsden Vector Calculus Solutions 5th Edition Manual PDF file for free Get. 4 Green’s Theorem 17. All vectors will be assumed proper unless otherwise stated. 5 Gauss Divergence. 2Vector Calculus. CS 189 Introduction to Machine Learning Spring 2020 Jonathan Shewchuk 1 Vector Calculus Review Proving. 3 The Fundamental Theorem for Line Integrals 17. • When dealing with large and/or time varying objects or phenomenon we must be able to describe the state at locations, and as a whole. Let x 2RN, y 2R , and z 2RK with y = f(x) and z = g(y), so we have the same. Professor. All of these have ﬁrst derivative equal to 0 at the origin, and thus it is natural to guess that (Df)(! 0)=! 0. Unless otherwise stated, every vector calculus result from this section is taken directly from the As with most denitions in vector calculus, this denition makes physical sense. Vector Calculus - GATE Study Material in PDF. Vector Calculus. IXL offers dozens of Calculus skills to explore and learn! Not sure where to start? Go to your personalized Recommendations wall to find a skill that looks interesting, or select a skill plan that aligns to your textbook, state standards, or standardized test. This text explains the theory of vector-valued functions, with detailed solutions of all It also reviews relevant topics in linear algebra and calculus of real-valued. Next PostNext. 15 Scalar Equation of a Plane a(x x 0) + b(y y 0) + c(z z 0) = 0 where (x 0;y 0;z 0) is a point on the plane and ha;b;ciis the vector normal to the plane. Home page url. •Recall: Fundamental Theorem of Calculus (FTC) •Definition: A vector field F is called a conservative vector field if there exist a potential, a function f, such that. Jamshidi Deﬁnition 5. A repository of tutorials and visualizations to help students learn Computer Science, Mathematics, Physics and Electrical Engineering basics. A vector ﬁeld is a function from IR2 to IR2 that typically represents some force. pdf: File Size: 1441 kb: File Type: pdf: Download File. Vector Calculus for Engineers covers both basic theory and applications. Keywords: vector calculus; mathematics for scientists/engineers; geometric interpre-tation. in Department of Electrical & Electronics Engineering BITS Pilani, Hyderbad Campus May 7, 2015 Vector Calculus EE208, School of Electronics Engineering, VIT. 1 The vector a = ae with a > 0. Edwards, Jr, Dover, 1995. 1—Intro to Parametric & Vector Calculus In Algebra, equations are graphed in two variables, x and y. Verify that this result holds when = x + y and V is the spherical volume x 2 +. The unique vector of length zero is denoted ~0 and satis es ~0 +~v= ~v. 2 Hyperboloids 12. The magnitude of vector is and vector is. Latest version (2021-01-02): VectorCalculus. Actually, we’ll see soon that eqn(5) plays a core role in matrix calculus. Vector Calculus Welcome,you are looking at books for reading, the Vector Calculus, you will able to read or download in Pdf or ePub books and notice some of author may have lock the live reading for some of country. The second, larger portion, teaches us the calculus of motion. Let us generalize these concepts by assigning n-squared numbers to a single point or n-cubed numbers to a single. UNIT – I - Chapter 2 Sections 13 to 21 & Chapter 3 Sections 24 to 31 of [1] UNIT – II - Chapter 4 Sections 35 to 42 of [1] UNIT – III - Chapter 1 Sections 11 , 12 & 13 of [2] UNIT – IV - Chapter 5 Sections 2. If you're interested in learning more about the history of vector calculus, Michael J. so that in a four-dimensional space, every vector is determined by the quadruple of the components (x 1,x 2,x 3,x 4). 13) The three numbers A i, i= 1;2;3, are called the (Cartesian) components of the vector A. We have aimed at presenting the broadest range of problems that you are likely to encounter—the old chestnuts, all the current standard types, and some not so standard. size) of the oject is clled sclr quntit, e. Tromba, Mznlnx, Aug 1, 2009, Education, 150 pages. Stating that the normal of a vector (x,y) is (-y,x) is repeating the question, and not a proof. Fourier Series & Harmonic Analysis. Eqn 20 is an extremely useful property in vector algebra and vector calculus applications. Introduction II. Advanced Engineering Mathematics, Kreyszig, 8th edition Less worked examples but covers the material thoroughly; applications oriented. Vector calculus with vector algebra , Paul McDougle, 1971, Mathematics, 608 pages. Matrix determinants and inverses. Many topics in the physical sciences can be analysed mathematically using the techniques of vector calculus. This text explains the theory of vector-valued functions, with detailed solutions of all It also reviews relevant topics in linear algebra and calculus of real-valued. 2 Line Integrals 17. vector_calculus. Vector Calculus - GATE Study Material in PDF. Download Vector Calculus Book PDF. Books selection & Best Mathematics Books play invaluable role to increase chances of success. 3B Vector Calculus Apply the divergence theorem to the vector eld u (x ) = a (x ) where a is an arbitrary constant vector and is a scalar eld, to show that Z V r dV = Z S d S ; where V is a volume bounded by the surface S and dS is the outward pointing surface element. F(x,y,z) = 1 p x2 +y2 +1 0 @ y x 1 1 A Unit vector ﬁeld. 9 The Divergence Theorem 2. is encoded in Ricci calculus by the -tensor i j. 13—Vector Calculus 2. zip Mathematics Handbook Volume1&2. University of Lethbridge has used APEX Calculus since 2015. The dot product of the vectors and is. 4MB, PDF) Download mirrors: Mirror 1. A rigorous and comprehensive introductory treatment, it features clear, readable proofs that illustrate the classical theorems of vector calculus, including the inverse and implicit function theorems. To do it properly, you have to use some linear algebra. Click Get Books and find your favorite books in the online library. Highlights of Calculus. 7 Vector Differentiation 1. Theory, examples, what you need to know, and for Fubini's Theorem. The MznLnx Exam Prep series is designed to help. So I wrote up this brief. If you're interested in learning more about the history of vector calculus, Michael J. 6 Gradients, Extreme Values, Diﬀerentials (Chapter 15). Page 4 Suppose, now, that < is a differentiable vector-valued function that maps an interval of real numbers c+ß,d into H'‚8. bra" and \geometric calculus," which have since become standard terms. Prerequisites Math 1B or equivalent. D Position Vector The position vector is the directed line segment OP from the origin of the coordinate system O to a generic point P. This note contains the following subcategories Vectors in R3, Cylinders and Quadric Surfaces, Partial Derivatives, Lagrange Multipliers, Triple Integrals, Line Integrals of Vector Fields , The Fundamental Theorem for Line Integrals ,Green’s Theorem , The Curl and Divergence. 6 that the quantity ˆvis a vector ﬁeld that represents the mass ﬂux. Home page url. This text is one of the first to treat vector calculus using differential forms in place of vector fields and other outdated techniques. Vector Calculus Marc Deisenroth Quantum Leap Africa African Institute for Mathematical Sciences, Rwanda Department of Computing Imperial College London @mpd37 [email protected] The supplement is intended for students who wish to gain a deeper un-. vector calculus - vector calculus Vector calculus is the study of vector elds and related scalar Chapter 16 Vector Calculus - San Dieguito Union High 16 Vector Calculus 16-1 16-5 Vector. calculus, and then covers the one-variable Taylor’s Theorem in detail. Lectures on Vector Calculus. This book covers calculus in two and three variables. Flap copy Miroslav Lovric, Ph. Dene a vector to be a directed line segment beginning at the origin, i. However, beginners report various difficulties dealing with the index notation due to its bulkiness. The goal is to show how vector calculus is used in applications. Differential Forms A Complement to Vector Calculus, Steven H. ½ º½ Î ØÓÖ. Multivariate Calculus; Fall 2013 S. Vector Calculus Independent Study Path Unit 8: Fundamental Theorems of Vector Calculus In single variable calculus, the fundamental theorem of calculus related the integral of the derivative of a function over an interval to the values of that function on the endpoints of the interval. Vector calculus Vector calculus (or vector analysis) is a branch of mathematics concerned with differentiation and integration of vector fields, primarily in 3 dimensional Euclidean space R3 Euclidean vector is a geometric object that has both a magnitude (or length) and direction. MULTIVARIABLEVECTOR-VALUEDFUNCTIONS 5-1. File Format: PDF/Adobe Acrobat. an integrated overview of Calculus and, for those who continue, a solid foundation for a rst year graduate course in Real Analysis. No need to wait for office hours or assignments to be graded to find out where you took a wrong turn. This paper. a vector and a scalar and can express the vector in component form, given an initial point and terminal point, and explain what the components describe. This text, Vector and Geometric Calculus, is intended for the second year vector calculus course. Download Vector Calculus Book PDF. This brief book presents an accessible treatment of multivariable calculus with an early emphasis on linear algebra as a tool. Calculus of vectors. The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial differentiation and multiple integration. It points from P1 to P2 and we can write P1 +~v. Eqn(5) is analogous to eqn(2), except the variable changes from a scalar to a vector. The chain rule can be extended to the vector case using Jacobian matrices. This is a non-profit website to share the knowledge. Vector Calculus by Miroslav Lovric, , available at Book Depository with free delivery worldwide. IA Vector Calculus. Tromba, [email protected] [Filename: errata_vc4e6. However, references are limited to. 1 0 x-2 0 2 y 0 2 4 z a b a+b 9. 19 Full PDFs related to this paper. The objectives in Advanced Calculus are typically to add depth to your first exposure to calculus. Graphical Educational content for Mathematics, Science, Computer Science. Examples: • Wind speeds and directions on the surface of the earth • Velocity of a moving fluid • Electromagnetic. There are four examples sheets for this course. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise. If n = 1, then !xi is the usual absolute value of x. Text: Vector Calculus, Linear Algebra, and Diﬀerential Forms 4th edition(2009), by J. 2 Graphical representation of the sum of two vectors: c = a+ b. (Also useful for JH SoCM) ML Boas, Mathematical Methods in the Physical Sciences, (Wiley). edu office: APM 5256, Office hours: M:3:30-4:30, W:4:30-5:30 Please show up at least 15 minutes before the end of the office hour. +# ) Vector Calculus! Read Rohlf, P576 to 577! Gradient Operator. The vector field W describes the velocity field for a rigid body rotation about the axis with angular speed equal to w 1 2 w 2 2 w 3 2. vector calculus - vector calculus Vector calculus is the study of vector elds and related scalar Vector Calculus - cs. Integrations in vector calculus Integration formulae Maxwell’s equations Maxwell’s equations – Faraday In the differential representation, ~—⇥E = ∂~B ∂t Performing the surface integration and applying Stokes’ theorem, Z ⌃ ( ~—⇥E)·dA = I ∂⌃ E ·d~s = ∂ ∂t Z ⌃ B·dA. Vector Calculus Marsden 6th Edition Pdf 11 Vector Calculus was written by and is associated to the ISBN: 9781429215084. Vector Calculus with Student Solutions Manual Set. This text explains the theory of vector-valued functions, with detailed solutions of all It also reviews relevant topics in linear algebra and calculus of real-valued. 2 DIFFERENTIAL LENGTH, AREA, AND VOLUME Differential elements in length, area, and volume are useful in. each further vector can be expressed as an multiple of these vectors. 2 Hyperboloids 12. THE GEOMETRY OF EUCLIDEAN SPACE Section 1. A velocity is a vector giving the speed and direction of motion. VECTOR CALCULUS. Books selection & Best Mathematics Books play invaluable role to increase chances of success. Other useful references concerning matrix calculus include [5] and [6]. Therefore, we can associate with f the function f2C() such that for each x2V, f(x) = P y˘x f(x;y)e xy and hence X() can be identi ed with C(). Visual Calculus is an easy-to-use Calculus grapher for Graphing limit, derivative function, integral VeryDOC PDF to Vector Converter (PDF2Vector) can be used to batch convert from Acrobat PDF. A vector field in the plane can be visualised as a collection of arrows with a given magnitude and direction, each attached to a point in the plane. MATH 20550: Calculus III Practice Exam 1 Multiple Choice Problems 1. Vector calculus is the essential mathematical tool for such analysis. •Theorem: If F is vector field defined on whose component functions have continuous partial derivatives and , then F is a conservative vector field. In Cartesian coordinates a = a 1e 1 +a 2e 2 +a 3e 3 = (a 1,a 2,a 3) Magnitude: |a| = p a2 1 +a2 2 +a2 3 The position vector r = (x,y,z) The dot. Dene a vector to be a directed line segment beginning at the origin, i. 3 Planes in 3-Space 12. Open Digital Education. single-variable calculus, and also relates to formulas for area and volume from MAT 169 that are de ned in terms of determinants, or equivalently, in terms of the dot product and cross product. tended to be used with the 6th Edition of our text Vector Calculus. CS 189 Introduction to Machine Learning Spring 2020 Jonathan Shewchuk 1 Vector Calculus Review Proving. This book presents the necessary linear algebra and then uses it as a framework upon which to build multivariable calculus. Recap of Vector Calculus S. Notable textbooks on vector calculus by Stewart and on electromagnetism by Purcell and Griffiths provide a much more thorough examination of these topics. Zinka [email protected] Created Date: 9/26/1997 11:56:08 AM. Tromba, [email protected] [Filename: errata_vc4e6. Any in-class announcement, verbal or written, is considered official addendum to this syllabus. 2 Vector Components and Dummy Indices Let Abe a vector in R3. Equality of Vectors Two vectors a and b are said to be equal written as a = b, if they have (i) same length (ii) the same or parallel support and (iii) the same sense. Vectors in Three Dimensions. Apply the divergence theorem to the vector eld u(x) = aφ(x) where a is an arbitrary constant vector and φ is a scalar eld, to show that. Distance Between Two Points; Circles. The graph of a function of two variables, say, z=f(x,y), lies in Euclidean space, which in the Cartesian coordinate system consists of all ordered triples of real numbers (a,b,c). Recap of Vector Calculus S. Therefore it need a FREE signup process to obtain the book. Introduction. The book's careful balance between theory, application, and historical development, provides readers with insights into how mathematics progresses and is in turn influenced by the natural world. scalar y column vector y 2Rm matrix Y 2Rm n scalarx @y @x row vector @y @x 2R m matrix @Y. Learn Vector Calculus online with courses like Vector Calculus for Engineers and Calculus through Data & Modelling: Vector Calculus. I teach two courses are in the undergraduate curriculum. Many quantities which are of interest in physics are both directed quantities (vectors) and can take on a continuous range of values, making calculus methods necessary. Unit vector corresponding to vector A is written as A with a bar on it. In the second formula, the transposed gradient (∇) is an n × 1 column vector, is a 1 × n row vector, and their product is an n × n matrix (or more precisely, a dyad); This may also be considered as the tensor product ⊗ of two vectors, or of a covector and a vector. vector calculus calculus of vector-valued functions Filling the Mobius strip. CS 189 Introduction to Machine Learning Spring 2020 Jonathan Shewchuk 1 Vector Calculus Review Proving. Temperature T is a scalar, and will certainly be a function of a position vector x = (x,y,z). Vector calculus with vector algebra , Paul McDougle, 1971, Mathematics, 608 pages. Partial diﬀerentiation. 9—Vector Calculus 1. This is why you remain in the best website to look the amazing books to have. Vector Calculus course. It is non-negative scalar. • To learn the vector calculus and its applications in engineering analysis • Expressions of vectors and vector functions • Refresh vector algebra • Dot and cross products of vectors and their physical meanings • To learn vector calculus with derivatives, gradient, divergence and curl • Application of vector calculus in engineering. Therefore, it is important to know calculus in many dimensions. Read an Excerpt. This establishes the rule for transforming any contravariant vector from one system to another. Vector Calculus by Miroslav Lovric, , available at Book Depository with free delivery worldwide. Vector Calculus IA. The goal is to show how vector calculus is used in applications. Notice that speed and velocity are not synonyms in physics -- the term velocity refers to a vector quantity and has both a magnitude (the speed) and direction. Vector Calculus. Download Notes as PDF. Introduction II. MR Spiegel, Vector Analysis, (Schaum, McGraw-Hill). Add Your Comments. 1 Change of Variables from Cartesian to. Vector calculus is a staple of the engineering disciplines. MAT 460 Vector Calculus SPRING 2017* *Important Note: All items on this syllabus are subject to change. Graphical user interfaces developed by the author which display three-dimensional surfaces, contour plots, gradient fields, and parametric curves and vector fields in both two and three dimensions will be presented with comments on their effectiveness. Vector Calculus – Study Materials. A vector x is then represented in the form. pdf 412 × 412; 18 KB. Java code samples from the book: calc3book_java. Multiple Integrals & Beta, Gamma Functions. 0 0 10 20 Figure3:Graphofs(t) Wenowwanttointroduceanewtypeoffunctionthatincludes,and. In Ricci calculus one distinguishes between free and bound indices. The proofs of most of the major results are either exercises or. khanacademy. Scalar and Vector fields A scalar field is one that has a single value associated with each point in the domain. Vector calculus is the essential mathematical tool for such analysis. Vector Calculus. 55 MB Format : PDF, ePub, Mobi Download : 454 Read : 359 Get This Book. 2 DIFFERENTIAL LENGTH, AREA, AND VOLUME Differential elements in length, area, and volume are useful in. Linear dependency of vectors Rem. This note contains the following subcategories Vectors in R3, Cylinders and Quadric Surfaces, Partial Derivatives, Lagrange Multipliers, Triple Integrals, Line Integrals of Vector Fields , The Fundamental Theorem for Line Integrals ,Green’s Theorem , The Curl and Divergence. Marsden, [email protected] Anthony J. Rent Vector Calculus 6th edition (978-1429215084) today, or search our site for other textbooks by Jerrold E. Fill Vector Calculus Book Pdf Download, Edit online. Notice that speed and velocity are not synonyms in physics -- the term velocity refers to a vector quantity and has both a magnitude (the speed) and direction. This textbook survival guide was created for the textbook: Vector Calculus, edition: 6. This edition features the exact same content as the traditional text in a convenient, three-hole- punched, loose-leaf version. Recap of Vector Calculus S. In these examples, b is a constant scalar, and B is a constant matrix. Vector Calculus. Rearranging the dot product formula to solve for gives us. Next: Introduction Up: Newtonhtml Previous: Further Investigation Vector Algebra and Vector Calculus Subsections. Vector Calculus with Student Solutions Manual Set. Vector Calculus - GATE Study Material in PDF In previous articles, we have already seen the basics of Calculus – Differentiation and Integration and applications. Download links and password may be in the description section, read description. Chapter 18. pdf to start downloading. 5Mb) Source (8Mb) Math 222 – Second Semester Calculus. If you find my work useful, please consider making a donation. Marsden and Anthony Tromba helps students gain an intuitive and solid understanding of calculus. Vector Calculus. 1—Intro to Parametric & Vector Calculus In Algebra, equations are graphed in two variables, x and y. (It includes more topics than Calculus. Traditionally we used PDF textbooks edited by Sean Fitzpatrick to align with the U of L calculus courses. i Math53Worksheets,7th Edition Preface This booklet contains the worksheets for Math 53, U. Graphical user interfaces developed by the author which display three-dimensional surfaces, contour plots, gradient fields, and parametric curves and vector fields in both two and three dimensions will be presented with comments on their effectiveness. Download Vector calculus marsden 6th edition pdf files Exam Prep for Vector Calculus by Marsden & Tromba, 5th Ed. In the previous sections, we have studied real-valued multivariable functions, that 250 chapter 5. For example, in one variable calculus, one approximates the graph of a function using a tangent line: 0 2 4-2 -1 1 2 x In the illustration above, the function g(x) = x2 is replaced by the simpler function ℓ(x) = 2x−1, a good approximation near the point x= 1. If Xis a continuous random variable with probability density function (PDF) f(x) = ce xfor x 0 and zero otherwise, (a) Find c(b) Find the cumulative distribution function F(x) (c) Find P(1 Vector Calculus VECTOR DIFFERENTIAL CALCULUS The vector differential calculus extends the VECTOR CALCULUS. outflow = inflow = constant; Geometrically, Divergence gives the rate at which the fluid entering in a rectangular parallelepiped per unit volume at the point. Vector Analysis and Cartesian Tensors, (Chapman and Hall). pdf from MATH 113 at Peking Uni. 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Notable textbooks on vector calculus by Stewart and on electromagnetism by Purcell and Griffiths provide a much more thorough examination of these topics. Vector Calculus – Study Materials. •“Vector Analysis and Cartesian Tensors”, Bourne and Kendall 1999 by Nelson. The divergence of a vector is written as $$abla \cdot {\bf v}$$, or $$v_{i,i}$$ in tensor notation. Scalar and Vector fields A scalar field is one that has a single value associated with each point in the domain. Vector Analysis and Cartesian Tensors, (Chapman and Hall). The book has the material typically covered in the third semester of a mainstream calculus course for science, mathematics, and engineering students. textbook-of-vector-calculus 2/4 Downloaded from cep. 5 Curl and Divergence 17. Download full Text Book Of Vector Calculus books PDF, EPUB, Tuebl, Textbook, Mobi or read online Text Book Of Vector Calculus anytime and anywhere on any device. •Theorem: Let C be a smooth curve given by. 4c Understand vector subtraction v – w as v + (–w), where (–w) is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. About the Author. The book's careful balance between theory, application, and historical development, provides readers with insights into how mathematics progresses and is in turn influenced by the natural world. dkarkada/coursenotes/Vector-Calculus. Enjoyed reading the book. It should be emphasized that this appendix cannot be seen as a textbook on vector algebra and analysis. Download it Vector Calculus books also available in PDF, EPUB, and Mobi Format for read it on your Kindle device, PC, phones or tablets. Marsden, [email protected] Anthony J. Includes index. Chapter 16 - Vector Calculus 16. Rethinking Calculus - Vector Calculus Notes - Prepared by Imad Ali with Contributions and Edits by This document is intended to provide a brief overview of the salient topics in Vector Calculus at. Differential Calculus -1: Determination of nth order derivatives of Standard functions – Problems. The vector differential calculus extends the basic c. Edwards, Jr, Dover, 1995. 48 KB ) Dancing Honeybee Using Vector Calculus to Communicate. Verify that this result holds when = x + y and V is the spherical volume x 2 +. Vector Calculus: Calculus, Edwards and Penney, 6th edition Accessible and colourful. Vector Calculus or Download Marsden Vector Calculus Solutions 5th Edition Manual PDF file for free Get. The graph of a function of two variables, say, z=f(x,y), lies in Euclidean space, which in the Cartesian coordinate system consists of all ordered triples of real numbers (a,b,c). This course is, normally known as , vector calculus, or multi variable calculus, or simply calculus-III. The proofs of most of the major results are either exercises or. The book's careful contemporary balance between theory, application, and historical development, provides readers with insights into how mathematics progresses and is in turn influenced by the natural world. in the last couple of videos we saw that we can describe a curve by a position vector-valued function and in very general terms it would be the x position as a function of time times the unit vector in the horizontal direction plus the Y position is a function of time times the unit vector in the vertical direction and this will essentially describe this though if you can imagine a particle. Vector Calculus Marc Deisenroth Quantum Leap Africa African Institute for Mathematical Sciences, Rwanda Department of Computing Imperial College London @mpd37 [email protected] The rela tion between the norm and the vector space structure of Rn is very important. 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/ hep-ex CERN-PH-EP-2015-160

Measurement of the centrality dependence of the charged-particle pseudorapidity distribution in proton--lead collisions at $\sqrt{s_{_{\rm{NN}}}} = 5.02$ TeV with the ATLAS detector

Pages: 28

Abstract: The centrality dependence of the mean charged-particle multiplicity as a function of pseudorapidity is measured in approximately 1 $\mu$b$^{-1}$ of proton--lead collisions at a nucleon--nucleon centre-of-mass energy of $\sqrt{s_{_{\rm{NN}}}} = 5.02$ TeV using the ATLAS detector at the Large Hadron Collider. Charged particles with absolute pseudorapidity less than 2.7 are reconstructed using the ATLAS pixel detector. The $p$+Pb collision centrality is characterised by the total transverse energy measured in the Pb-going direction of the forward calorimeter. The charged-particle pseudorapidity distributions are found to vary strongly with centrality, with an increasing asymmetry between the proton-going and Pb-going directions as the collisions become more central. Three different estimations of the number of nucleons participating in the $p$+Pb collision have been carried out using the Glauber model as well as two Glauber--Gribov inspired extensions to the Glauber model. Charged-particle multiplicities per participant pair are found to vary differently for these three models, highlighting the importance of including colour fluctuations in nucleon--nucleon collisions in the modelling of the initial state of $p$+Pb collisions.

Note: 28 pages plus author list + cover pages (46 pages total), 14 figures, 4 tables, submitted to EPJC, All figures including auxiliary figures are available at https://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/PAPERS/HION-2012-15/

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Answer c is correct because c have 3 alpha hydrogen while d have only 2.

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The thermodynamics are always the same. If you go from carbon monoxide to methane, there is only one final enthalpy, and if you have the choice of going to methane or methanol the thermodynamically preferred product is always the same. The difference indeed less within the different pathways and the activation barriers needed to get from one intermediate to ...

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The terminal alkyne has acidic $\ce{H}$ and therefore will give acid base reaction with $\ce{NaNH3}$ and go $\ce{NH4+}$ and $\ce{RC#C- Na+}.$

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THF is a polar aprotic solvent, that is, it can stabilize positive charges but not negative ones since it won't form hydrogen bonds with the help of any of its own hydrogen atoms. Thus the carbocationic intermediate, that is the mercurinium ion is still stabilized, allowing the reaction to proceed. And in Hydroboration, THF acts as the solvent as well but ...

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text:
# How the encoding works

Given a list of bits:

• Hold a prime (starting with 2)
• Have a list
• For each bit in the input
• If it's the same as the previous bit, add the prime you're holding to the list
• If it's different, hold the next prime and add that to the list
• Return the product of all the numbers in your list
• For the first bit, assume the previous bit was 0

Note: these steps are for illustration purposes only, you are not required to follow them.

# Examples

Input: 001
hold 2

0:         add 2 to the list
0:         add 2 to the list
1: hold 3, add 3 to the list

list: 2,2,3
Output: 12

Input: 1101
hold 2

1: hold 3, add 3 to the list
1:         add 3 to the list
0: hold 5, add 5 to the list
1: hold 7, add 7 to the list

list: 3,3,5,7
Output: 315

### Some more examples:

000000000 -> 512
111111111 -> 19683
010101010 -> 223092870
101010101 -> 3234846615
011101101 -> 1891890
000101101010010000 -> 3847834029582062520

# Challenge

Write an encoder and a decoder for this encoding method.

(The decoder reverses the process of the encoder).

# Input / Output

• The encoder can take input in any reasonable format

• The encoder must output either an integer or a string

• The decoder must take input in the same format that the encoder utputs

• The decoder must output the same format that the encoder takes as input

In other words decoder( encoder( input ) ) === input

### Notes

• The decoder may assume that its input is decodable
• Your answer only has to deal with integers that your language can natively support without using (long, bigInt, etc.), be reasonable, if you language only supports ints up to 1, maybe reconsider posting an answer

# Scoring

Your score is the sum of the lengths in bytes of the encoder and decoder.

If you need to import a module, the import can be counted only once provided that your encoder and decoder can coexist in the same file and be reused (like functions).

Default loopholes are forbidden.

This is so the shortest score for every language wins.

• Is that last example mandatory, or can we limit the output to up to 64 bits (2^63-1 / 9223372036854775808)? – Kevin Cruijssen Apr 26 '18 at 12:03
• @KevinCruijssen No, your answer only has to work for integers that your language can handle. – Asone Tuhid Apr 26 '18 at 12:06
• @KevinCruijssen *handle natively without libraries of bigints, i'll clarify – Asone Tuhid Apr 26 '18 at 13:07

# 05AB1E, 13 bytes

## Encoder, 8 bytes

0ì¥ĀηOØP

Try it online!

Explanation

0ì          # prepend 0 to input
¥         # calculate deltas
Ā        # truthify each
η       # calculate prefixes
O      # sum each
Ø     # get the prime at that index
P    # product

## Decoder, 5 bytes

Ò.ØÉJ

Try it online!

Explanation

Ò       # get prime factors of input
.Ø     # get their indices among the primes
É    # check for oddness
J   # join

# Jelly, 17 bytes

### Encoder (10 bytes):

0;IA+\‘ÆNP

Try it online!

### Decoder (7 bytes):

ÆEĖŒṙḂ¬

Try it online!

### How?

Encoder:

0;IA+\‘ÆNP - Link: list of integers (1s and 0s)  e.g. [1,1,1,1,0]
0;         - prepend a zero                           [0,1,1,1,1,0]
I        - incremental differences                  [1,0,0,0,-1]
A       - absolute values                          [1,0,0,0,1]
+\     - cumulative reduce with addition          [1,1,1,1,2]
‘    - increment each of the results            [2,2,2,2,3]
ÆN  - get the nth prime for each result        [3,3,3,3,5]
P - product of the list                      405

Decoder:

ÆEĖŒṙḂ¬ - Link: integer         e.g. 405
ÆE      - prime exponent array       [0,4,1] (representing 2^0*3^4*5^1)
Ė     - enumerate                  [[1,0],[2,4],[3,1]]
Œṙ   - run-length decode          [2,2,2,2,3]
Ḃ  - bit (mod 2)                [0,0,0,0,1]
¬ - logical NOT                [1,1,1,1,0]

# JavaScript (ES6), 130 bytes

I/O: array of bits ↔ integer

## Encoder, 71 bytes

a=>a.map(p=k=>r*=(g=i=>n%--i?g(i):i<2?n:g(++n))(n+=p^(p=k)),r=1,n=2)&&r

Try it online!

## Decoder, 59 bytes

D=(n,k=2,x=b=0)=>k>n?[]:n%k?D(n,k+1,1):[b^=x,...D(n/k,k,0)]

Try it online!

# Java 10, 209 bytes

## Encoder, 124 bytes

s->{long p=48,P=2,r=1,n,i;for(int c:s.getBytes()){if(p!=(p=c))for(n=0;n<2;)for(n=++P,i=2;i<n;n=n%i++<1?0:n);r*=P;}return r;}

Try it online.

Explanation:

s->{                // Method with String parameter and long return-type
long p=48,        //  Previous character, starting at '0'
P=2,         //  Current prime, starting at 2
r=1,         //  Result, starting at 1
n,i;         //  Temp-integers
for(int c:s.getBytes()){
//  Loop over the digits of the input-String as bytes
if(p!=(p=c))    //   If the current and previous digits are different
for(n=0;      //    Reset n to 0
n<2;)     //    And loop as long as n is still 0 or 1
for(n=++P,  //     Increase P by 1 first with ++P, and set n to this new P
i=2;i<n;n=n%i++<1?0:n);
//     Check of the current n is a prime
//     If it remains the same it's a prime, if it becomes 0 or 1 not
r*=P;}          //   Multiply the result by the current prime P
return r;}        //  Return the result

## Decoder, 85 bytes

n->{var r="";for(long P=2,f=0,i=1;++i<=n;)for(;n%i<1;n/=P=i)r+=i!=P?f^=1:f;return r;}

Try it online.

Explanation:

n->{                // Method with long parameter and String return-type
var r="";         //  Result-String, starting empty
for(long P=2,     //  Current prime, starting at 2
f=0,          //  Flag integer, starting at 0
i=1;++i<=n;)  //  Loop i in the range [2,n]
for(;n%i<1;     //   Inner loop over the prime factors of n
n/=P=i)     //     After every iteration: divide n by i,
//     and set P to i at the same time
r+=i!=P?      //    If i and P are not the same
f^=1      //     Append the opposite of the flag f (0→1; 1→0)
:          //    Else:
f;        //     Append the flag f
return r;}        //  Return the result

• You can save 2 bytes by changing long to int. – Asone Tuhid Apr 26 '18 at 19:15

# Husk, 18 bytes

### Encoder, 11 bytes

Πmo!İp→∫Ẋ≠Θ

Try it online!

### Decoder, 7 bytes

mȯ¬%2ṗp

Try it online!

### How they work

Encoder:

Πmo!İp→∫Ẋ≠Θ – Full program. Takes input from the first CLA, outputs to STDOUT.
Θ – Prepend a default element (0 in this case).
Ẋ≠  – Map over pairs of adjacent elements with ≠ (not equals). In Husk,
some operators return other useful things than just boolean values.
Here, ≠ returns the absolute difference between its two arguments.
∫    – Cumulative sums.
mo         – Map the following function over the list of sums:
İp        – Yield the infinite list of primes.
!  →       – And index into it with the current sum incremented.
Π           – Take the product.

Decoder:

mȯ¬%2ṗp – Full program.
p – Prime factorization.
mȯ      – Map the following function over the list of factors:
ṗ    – Retrieve the index in  the list of primes.
%2     – Modulo 2.
¬       – Logical NOT.

# Python 2, 234193 174 bytes

## Encoder, 116101 97 bytes:

Uses Wilson's theorem.

i=input()
P=n=x=r=1
while i:
P*=n*n;n+=1
if P%n:t=(i+[x]).index(x);i=i[t:];r*=n**t;x^=1
print r

Try it online!

## Decoder, 11892 77 bytes:

i=input()
r=[]
n=x=1
while~-i:
n+=1;x^=i%n<1
while i%n<1:r+=x,;i/=n
print r

Try it online!

# J, 34 bytes

Heavily inspired by Jonathan Allan's Jelly solution!

## Encoder: 23 bytes

[:*/[:p:[:+/\2|@-~/\0,]

Try it online!

                    0,]  NB. prepend 0 to the input
2  -~/\     NB. find the differences
|@         NB. and their absolute values
[:+/\            NB. running sums
[:p:                 NB. n-th prime
[:*/                     NB. product

I don't like those many cap forks [: - it should be golfable.

## Decoder: 11 bytes

2|[:_1&p:q:

Try it online!

        q:    NB. prime factorization
[:_1&p:      NB. find the number of primes smaller than n
2|             NB. modulo 2

# Jelly, 15 bytes

### Encoder, 9 bytes

0;ŒgL€ÆẸH

Try it online!

### Decoder, 6 bytes

Æf’ÆCḂ

Try it online!

# C (gcc), 180 184 bytes

• Added four bytes so that output formats match.

# 102 bytes -- Encoder

p,r,i,m;e(char*_){for(m=0,p=1,i=2;*_;m=*_++-48,p*=i)if(*_-48-m)for(i++,r=2;r<i;i%r++||(r=2,i++));i=p;}

Try it online!

# 82 bytes -- Decoder

d(C){for(m=C%2,r=2+m,p=2;C>1;p++)if(C%p<1)p-r&&(m=!m,r=p),putchar(m+48),C/=p,p=1;}

Try it online!

• @AsoneTuhid Sorry, misread. – Jonathan Frech Apr 26 '18 at 11:31
• @AsoneTuhid Now added a decoder. Hopefully compliant now. – Jonathan Frech Apr 26 '18 at 11:53
• @ovs True; thanks for your remark. – Jonathan Frech Apr 26 '18 at 14:21

# Gol><>, 29 + 39 = 68 bytes

## Encoder, 29 bytes

021IEh{$:}-Q$TP:SP?!t$|1k*3R!  Try it online! ## Decoder, 39 bytes 02I:MZ;:2k%:z}Q$TP:SP?!t$|1k,{{-z:N}3R!  Try it online! ### How these work Encoder 021IEh{$:}-Q$TP:SP?!t$|1k*3R!

021                            Setup the stack as [last bit, current prime, current output]
IEh                         Take input as int; if EOF, print top as number and halt
{$:}-Q | If the next bit is different from the last bit...$                    Move the prime to the top
T      t            Loop indefinitely...
P:SP?!               Increment; if prime, skip t i.e. break
$Move the prime to the correct position 1k* Multiply the prime to the output 3R! Skip 3 next commands (the init part) Loop the entire program until EOF --- Decoder 02I:MZ;:2k%:z}Q$TP:SP?!t\$|1k,{{-z:N}3R!

02I                  Setup the stack as [last bit, current prime, encoded]
:MZ;              If encoded == 1, halt
:2k%          Compute encoded modulo prime
:z}       Store NOT of the last at the bottom of the stack
Q...|  Same as encoder's next-prime loop
1k,                  Divide encoded by prime (assume it is divisible)
{{                Pull out the two bits at the bottom
-z              Compute the next bit
:N}           Print as number with newline, and move to the bottom
3R!        Skip 3 init commands
Loop the entire program until finished

It'd be better if I could golf down the next-prime loop...


text:
# Topology and geometry: extremal and typical

## A Zoom seminar

This is an online seminar for the COVID era organized by Fedya Manin and Shmuel Weinberger. It will focus on whatever interests us, but mainly quantitative questions in geometry and topology.

Zoom links are distributed via a mailing list. Please sign up if you would like to attend.

Confirmed speakers include:

## Schedule and abstracts

The seminar will run on Mondays, roughly biweekly. Unless otherwise specified, the talks will be at noon Eastern Time, which (for now) is:
• 11am in Chicago
• 9am in Los Angeles
• 5pm in London
• 6pm in Paris
• 7pm in Moscow and Tel-Aviv
Our apologies to friends in Australia and East Asia for whom this is a terrible time.

Expand the items in this list to see abstracts.

August 17, 2020:
Robert Young (NYU)

How do you build a complicated surface? How can you decompose a surface into simple pieces? Understanding how to construct an object can help you understand how to break it down. In this talk, we will present some constructions and decompositions of surfaces based on uniform rectifiability. We will use these decompositions to study problems in geometric measure theory and metric geometry, such as how to measure the nonorientability of a surface and how to optimize an embedding of the Heisenberg group into L1 (joint with Assaf Naor).

August 31, 2020:
Panos Papasoglu (Oxford)

The Uryson width of an n-manifold gives a way to describe how closely it resembles an $(n-1)$-dimensional complex. It turns out that this is a useful tool to approach several geometric problems.

In this talk we will give a brief survey of some questions in ‘curvature-free’ geometry and sketch a novel approach to the classical systolic inequality of Gromov. Our approach follows up recent work of Guth relating Uryson width and local volume growth. For example we deduce also the following result of Guth: there is an $\epsilon_n>0$ such that for any $R>0$ and any compact aspherical n-manifold M there is a ball B(R)$of radius R in the universal cover of M such that$\operatorname{vol}(B(R))\geq \epsilon_n R^n$. September 14, 2020: Mark Pengitore (Ohio State) In this talk, we will relate homological filling functions and the existence of coarse embeddings. In particular, we will demonstrate that a coarse embedding of a group into a group of geometric dimension 2 induces an inequality on homological Dehn functions in dimension 2. As an application of this, we are able to show that if a finitely presented group coarsely embeds into a hyperbolic group of geometric dimension 2, then it is hyperbolic. Another application is a characterization of subgroups of groups with quadratic Dehn function. If there is enough time, we will talk about various higher dimensional generalizations of our main result. September 21, 2020 (note odd week): Alexey Balitskiy (MIT) The Urysohn d-width of a metric space quantifies how well it can be approximated by a d-dimensional simplicial complex. We discuss various questions of the following flavor: how does knowledge of the width of certain pieces of a riemannian manifold help us to estimate the total width of the whole manifold? Here are two examples. 1. A waist-type inequality: If the euclidean 3-ball is sliced into a 1-parametric family of (possibly singular) surfaces with$\operatorname{rank} H_1 \le b$then at least one of them has 1-width at least ~1/b (so it's "essentially 2-dimensional"). 2. The width can behave counterintuitively: it can happen that an n-manifold has substantial$(n-1)\$-width but all its unit balls are "almost 1-dimensional" (that is, of small 1-width).
Based on joint work with Sasha Berdnikov.

October 12, 2020:
Rina Rotman (Toronto), TBA

Add this Google Calendar if you'd like:


text:
# Nuclear power

.

The difference between genius and stupidity is that genius has its limits.

— Albert Einstein

.

Never underestimate the power of human stupidity.

— Robert A. Heinlein

.

Stupidity is like nuclear power; it can be used for good or evil.

— Dilbert

.

.

.

2009.07.07 Tuesday $ACHK$


text:
# AM-GM inequality in a specific question

If $k_1$ and $k_2$ are the least values of the functions $f(x)=2log_{10}(x)-log_x(0.01)$ and $g(x)=e^x+e^{-x}$.Then what is the value of $\frac{k_1}{k_2}$?

I found the solution as :-

This uses the concept that A.M is greater than or equal to G.M. But the problem is I do not know how to apply this concept and how has that been applied in the answer to this question. I'd be grateful if you explain me this concept.

• – StubbornAtom Oct 11 '16 at 7:52

If I understand your question correctly, you're asking how they get $f(x) \geq 4$ using AM-GM. AM-GM states that for non-negative real numbers $a,b$ $\frac{a+b}{2} \geq \sqrt{ab}$
$\frac{2\log x}{\log 10} + \frac{2\log 10}{\log x} \geq 2\sqrt{\frac{2\log x}{\log 10}\frac{2\log 10}{\log x}} = 2 \cdot 2$


text:
EX0: BPP-completeness

1.

L ≝ \lbrace (M, x, 1^t) \mid M \text{ accepts on input } x\text{ in time at most }t \rbrace

is $NP$-complete

$∈ NP$

1. check that the input is of the form $(M, x, 1^t)$
2. execute $M$ on $x$
3. … while at the same time keeping a timeout

EX3: The BP-operator

$B ≼_r C$

iff $∃ M; ∀x, ℙ(B(M(x)) = C(x)) ≥ \frac 2 3$

$L ∈ BP \cdot 𝒞$:

iff $∃ M \text{ and } D ∈ 𝒞$ s.t.

• $ℙ(M(x, r) ∈ D) ≥ \frac 2 3$ if $x∈L$
• $ℙ(M(x, r) ∉ D) ≥ \frac 2 3$ if $x∉L$

1.

Show that $\lbrace L \mid L ≼_r L' \text{ for } L'∈𝒞 \rbrace = BP \cdot 𝒞$

Let $L$ s.t. $∃L’∈ 𝒞$ s.t. $L ≼_r L’$

There exists $M$ s.t. for all $x$,

p_x ≝ ℙ(L'(M(x, r)) = L(x)) ≥ \frac 2 3
• $p_x = ℙ(M(x, r) ∈ L’) ≥ \frac 2 3$ if $x∈L$
• $p_x = ℙ(M(x, r) ∉ L’) ≥ \frac 2 3$ if $x∉L$

2. BPP is closed under polynomial-time randomized reduction

If $𝒞 ∈ BPP$, and $𝒞’ ≼ 𝒞$, then $𝒞’ ∈ BPP$

ℙ(𝒞(M(x, r)) = 𝒞'(x)) ≥ \frac 2 3

Let $M_{𝒞, ε}$ be the PTM recognizing $𝒞$ with proba $≥ε$

Let $x ∈ Σ^\ast_{𝒞’}$. We run

M_{𝒞, ε}(M(x, r_1), r_2)

$x ∉ 𝒞’$: $p_ε = ℙ(M_{𝒞, ε}(M(x, r_1), r_2) = 1) ≤ \frac 1 3 + ε$

Let $ε ≝ \frac 1 {12}$, then $% $

3.

Composition of reductions.

Suppose $A ≼_r B$ and $B ≼_r C$.

Let $M_1, M_2$, $r_1, r_2$ given by the definition.

• $ℙ(C(M_1(x, r)) ≠ B(x)) ≤ ε_1$
• $ℙ(B(M_1(x, r)) ≠ A(x)) ≤ ε_2$

If $x ∈ L_A$,

ℙ(B(M_1(x, r)) = 0) ≤ ε_2 + ε_1

EX 4: The class $BP \cdot NP$

1. $BP \cdot P = BPP$

$L ∈ BP \cdot P$:

iff $∃ M \text{ PTM and } D ∈ P$ s.t.

• $ℙ(M(x, r) ∈ D) ≥ \frac 2 3$ if $x∈L$
• $ℙ(M(x, r) ∉ D) ≥ \frac 2 3$ if $x∉L$

i.e $L ∈ BPP$

2. $BP \cdot NP = AM$

$⊆$

If $L ∈ \textbf{AM}$. Let $(A, M, D)$ given by the definition.

Let $D' = \lbrace x \mid ∃y. x \# y ∈ D\rbrace$

NB: It is implicitly assumed that $\vert y \vert$ is polynomial in $\vert x \vert$.

$D’ ∈ NP$ by definition of $NP$ by certificate.

• if $x ∈ L$: $ℙ(A(x, r) ∈ D') = ℙ(∃y. A(x, r) \# y ∈ D) \\ ≥ ℙ(A(x, r) \# M(A(x, r))) ≥ 1 - \frac 1 {2^{n^l}}$

• if $x ∉ L$: $ℙ(A(x, r) ∉ D') = ℙ(∀y. A(x, r) \# y ∉ D) ≤ 1 - \frac 1 {2^{n^l}}$

4.

$BPP ⊆ Σ_2$

If $x∈L$ ⟺

∃ t_0, ⋯, t_{q(n)/r}; ∀r, \bigvee_i M(x, r \oplus t_i) = \top
∀n, ∃r_n, ∀x \text{ of len } n, M(x, r_n) \text{ not wrong}

Tags:

Updated:


text:
# Is $\log^{*}(\log(n)) = \Theta (\log(\log^{*}(n)))$?

Which one is asymptotically larger? $$\log^*(\log(n))$$ or $$\log(\log^*(n))$$? I think they are asymptotically tight bounds for each other (one is $$\Theta$$ of the other).

Suppose that $$\log^* n = k$$. This means it takes $$k$$ many $$\log$$'s to reduce $$n$$ below some constant. Therefore $$\log^* \log n = k-1$$ (assuming $$k$$ is not too small), whereas $$\log \log^* n = \log k$$.
Suppose $$f_1(n)=\log \log^{*}(n)$$, and $$f_{2}(n)=\log^{*} \log(n)$$.
Let $$n=2^m$$ then According the definition of iterated function:
$$\log^* n$$ (usually read "log star"), is the number of times the logarithm function must be iteratively applied before the result is less than or equal to 1.
So $$f_1(2^m)=\log \log^*(2^m)=\log \log^*\log(2^m)+1=\log\log^*m+1=\theta(\log\log^*m)$$
and $$f_2(2^m)=\log^*\log(2^m)=\log^*(m)+1=\theta(\log^*m)$$ Finally, for sufficient large $$n$$ (i.e. as $$n\to \infty$$) we conclude that: $$f_1(2^m)=O(f_2(2^m)).$$ Because let $$\log^*(m)=k$$ then: $$f_1(2^m)=\log k$$ and $$f_2(2^m)=k.$$ Obviously order of growth two mentioned function follow below relation $$f_1(2^m)=O(f_2(2^m)).$$


text:
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Concept

# Inverse Operations

Inverse operations are two operations that, all else being equal, undo one another. For instance, adding $6$ and subtracting $6$ are inverse operations because they cancel each other out. \begin{aligned} &x+6\\ &x+6{\color{#0000FF}{\ -\ 6}}\\ &x \end{aligned} In an expression like $x+6,$ the addition of $6$ to $x$ is eliminated by performing the inverse operation: a subtraction of $6.$ Using inverse operations on an equation, however, is a little different: in order to adhere to the Properties of Equality, any operation performed on one side of an equation must also be performed on the other side to maintain equality. \begin{aligned} x\div{2}&=1\\ x\div{2}{\color{#0000FF}{\ \times{\ 2}}}&=1{\color{#0000FF}{\ \times{\ 2}}}\\ x&=2 \end{aligned} In this case, the division by $2$ on one side of the equation could only be eliminated by a multiplication by $2$ on both sides of the equation. The result of applying the Properties of Equality on an equation is an equivalent equation.


text:
I have a summation of the form:

$$\sum_{m\in\mathbb{Z}}m\exp\left\{-\frac{B}{2}\left(z-\frac{2\pi iv}{B}-\frac{2\pi m}{BL_2}\right)^2\right\}$$ where $B, v, L_2$ are constants and $z$ is a variable. I've been trying to look for a solution via Gauss sums, but have found anything of help yet (my background is in theoretical physics so I have to read and learn number theory as I go). I was maybe thinking about using a Fourier expansion of a function which would produce the series above, but haven't found anything yet...

Any suggestions?

This is related to a derivative of a Jacobi theta-function: if we write $$\theta(z,\tau) = \sum_{m = -\infty}^{\infty} \exp{(\pi i m^2 \tau + 2\pi i n z)},$$ then $$\sum_{m = -\infty}^{\infty} \exp{\left(-\frac{B}{2}(z-2\pi i \nu/B)^2+\frac{2\pi m}{L_2}(z-2\pi i \nu/B) - \frac{2\pi^2}{BL_2^2}m^2 \right)} = e^{-B(z-2\pi i \nu/B)^2/2} \theta\left(\frac{z-2\pi i \nu/B}{iL_2},\frac{2\pi i}{BL_2^2}\right),$$ and then your sum is $$e^{-B(z-2\pi i \nu/B)^2/2} \frac{L_2}{2\pi}\frac{\partial}{\partial z}\theta\left(-i\frac{z-2\pi i \nu/B}{L_2},\frac{2\pi i}{BL_2^2}\right)$$ (Notice that the derivative brings down a $2\pi m/L_2$ inside the sum.) There is in general no way to write this more simply.
• No, I don't think so. The logarithmic derivatives of the theta-functions are given in dlmf.nist.gov/20.5#ii , and are decidedly not constant. Quotients of theta-functions satisfy nonlinear first-order differential equations, and there is of course the heat equation relating the second $z$-derivative to the $\tau$-derivative, but I think those are the only ones possible (note the derivative of a theta-function has additive quasi-periodicity, while the function itself is multiplicatively quasi-periodic). – Chappers May 19 '17 at 13:09


text:
MathSciNet bibliographic data MR205952 10.21 (14.49) Ihara, Yasutaka Discrete subgroups of ${\rm PL}(2,\,k\sb{\wp })$${\rm PL}(2,\,k\sb{\wp })$. 1966 Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) pp. 272–278 Amer. Math. Soc., Providence, R.I. Links to the journal or article are not yet available

For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews.


text:
# Generalized least squares with insignificant predictor variable

Suppose I have fitted an standard linear regression mode $Y=\beta_0+\beta_1X_1+\beta_2X_2+\beta_3X_3+\epsilon$. Based on the ACF plot or PACF plot of the residuals for this regression model, I found that the $\epsilon$'s are serially auto-correlated that can be modeled as an AR(1) i.e. $\epsilon=\phi\epsilon_{t-1}+\nu_t$, where $\nu_t$ are white noise. I decided to fit the above model with Generalized least squares approach with the following code

library(nlme)
model.gls=gls(Y~X_1+X_2+X_3,data=dat, correlation=corARMA(p=1), method="ML")

A part of output in R is as follows:

Correlation Structure: AR(1)
Formula: ~1
Parameter estimate(s):
Phi
0.8067566

Coefficients:
Value  Std.Error    t-value p-value
(Intercept) -3.172158 0.01504180 -210.88950  0.0000
x1          -3.890222 0.14141947  -27.50839  0.0000
x2           0.089368 0.11667430    0.76596  0.4456
x3          -0.155348 0.10190694   -1.52441  0.1308

Clearly $X_2$ and $X_3$ are not significant based on the t-test.

Q1: Is it a correct approach to drop $X_2$ and $X_3$ and refit a new Generalized least squares model with the same correlation structure (i.e. AR(1))? In this case the code should be like:

model.gls=gls(Y~X_1,data=dat, correlation=corARMA(p=1), method="ML")

I am asking this because the AR(1) has been selected while all $X_1$, $X_2$ and $X_3$ have been included in the linear regression model. So dropping two terms may change the correlation structure (i.e AR(1)) to another ARMA model.

If the approach in Q1 is not correct, how can we proceed and handle the non-significant terms, $X_2$ and $X_3$ in the fitted Generalized least squares?

• There is no reason to drop "insignificant" terms, and it will create a bias to do so. – Frank Harrell Dec 19 '12 at 22:06
• Can you provide any reference to support your comment? – Stat Dec 19 '12 at 22:13
• @FrankHarrell - isnt that the whole idea of dropping insignificant terms? You trade increased bias for decreased variance. – probabilityislogic Dec 19 '12 at 23:10
• Do you have an a apriori reason for believing that you have AR(1) error? Is Y a time series, where you might expect a carry-over from one observation to the next? If so, continue to model it. Other point: the p-values of individual coefficients can mislead -- not always the best way to do variable selection. Either one of X2 or X3 might be significant "were other dear charmer away." – Placidia Dec 20 '12 at 1:11
• I checked the ACF and PACF of the residuals in the linear model and AR(1) seems to be a good model for that. Yes, $Y$ is a time series. What do you mean by "continue to model it"? I think my questions are clear ... – Stat Dec 20 '12 at 1:28


text:
# Does the sum $\sum_{n \geq 1} \frac{2^n\operatorname{mod} n}{n^2}$ converge?

I am somewhat a noob, and I don't recall my math preparation from college. I know that the sum $\sum_{n \geq 1} \frac{1}{n}$ is divergent and my question is if the sum $$\sum_{n \geq 1} \frac{2^n\operatorname{mod} n}{n^2}$$ converges. I think is not but I do not know how to prove that! Thanks!

• you can use Mathjax when typing . – Nizar Oct 12 '16 at 9:30
• Intuitively it diverges, as $2^n\bmod n$ must be $O(n)$, tightly. – Yves Daoust Oct 12 '16 at 9:38
• @YvesDaoust Could you elaborate more on why $2^n\mod n$ is $\mathcal{O}(n)$? – Jacky Chong Oct 12 '16 at 9:40
• My thoughts exactly I think it is the same thing with sum_{n>=1} \frac{1}{2n} as n/2 is the average of 2^n mod n – Aurelian Florea Oct 12 '16 at 9:43
• @JackyChong I expect $2^n\bmod n$ to be uniformly distributed in $[0,n-1]$. (Of course anything $\bmod n$ is $O(n)$, so the point is more about the "expectation" of $2^n\bmod n$). – Yves Daoust Oct 12 '16 at 9:44

In this answer, we prove that

$$\sum_{n=1}^{\infty} \frac{2^n \text{ mod } n}{n^2} = \infty. \tag{*}$$

Idea. The intuition on $\text{(*)}$ comes from the belief that the sequence $(2^n \text{ mod } n)/n$ is equidistributed on $[0, 1]$, which is quite well supported by numerical computation.

Proving this seems quite daunting, though, so we focus on a particular subsequence which is still good to give a diverging lower bound of $\text{(*)}$. To be precise, we focus on the indices of the form $n = 5p$ for some prime $p$ and prove that the corresponding sum is comparable to the harmonic series for primes $\sum_p \frac{1}{p}$, which also diverges.

Proof. To this end, we consider the sequence $(a_k : k \geq 0)$ in $[0, 1)$ defined by

$$a_k = \left\{ \frac{2^{5p_k}}{5p_k} \right\},$$

where $\{ x \} = x - \lfloor x \rfloor$ is the fractional part of $x$ and $p_k$ is the $k$-th prime number. Now focusing only on the index $n = 5p_k$ for some $k$, we can bound the sum $\text{(*)}$ below by

$$\sum_{n=1}^{\infty} \frac{2^n \text{ mod } n}{n^2} = \sum_{n=1}^{\infty} \frac{1}{n}\left\{ \frac{2^n}{n} \right\} \geq \sum_{k=1}^{\infty} \frac{a_k}{5p_k}.$$

So it suffices to prove that this bound diverges. First, for any prime $p$ we have

$$2^{5p} \equiv 2^5 \equiv 32 \pmod{p}.$$

This allows us to write $2^{5p} = mp + 32$ for some non-negative $m$. Next, notice that any prime $p$ other than $2$ and $5$ are either of the form $p = 4k+1$ or of the form $p = 4k+3$. Depending on which class $p$ falls in, we find that

$$2^{5p} \equiv 2^p \equiv \begin{cases} 2, & \text{if } p =4k+1 \\ 3, & \text{if } p =4k+3 \end{cases} \pmod{5}.$$

What this tells about $m$ is as follows:

$$m \equiv \begin{cases} 0, & \text{if } p =4k+1 \\ p^{-1}, & \text{if } p =4k+3 \end{cases} \pmod{5}.$$

(Here, $p^{-1}$ is the multiplicative inverse of $p$ modulo $5$.) From this, for $p_k > 32$ we have the following estimate:

$$a_k \geq \frac{1}{5} \quad \text{if } p_k \equiv 3 \pmod{4}.$$

Consequently, by the PNT for arithmetic progression,

$$\frac{a_1 + \cdots + a_n}{n} \geq \frac{1}{5} \frac{\pi_{4,3}(p_n) + \mathcal{O}(1)}{\pi(p_n)} \xrightarrow[n\to\infty]{} \frac{1}{10}.$$

(The $\mathcal{O}(1)$-term appears by discarding terms with $p_k \leq 32$.) Finally, let $s_n = a_1 + \cdots + a_n$. Then by summation by parts, as $N \to \infty$ we have

\begin{align*} \sum_{k=1}^{N} \frac{a_k}{5p_k} &= \frac{1}{5} \bigg( \frac{s_N}{p_N} + \sum_{k=1}^{N-1} \left( \frac{1}{p_k} - \frac{1}{p_{k+1}} \right) s_k \bigg) \\ &\geq \frac{1}{5} \sum_{k=1}^{N-1} \left( \frac{1}{p_k} - \frac{1}{p_{k+1}} \right) \frac{k}{11} + \mathcal{O}(1) \\ &\geq \frac{1}{55} \sum_{k=1}^{N} \frac{1}{p_k} + \mathcal{O}(1). \end{align*}

Taking $N \to \infty$, this series diverges by the harmonic series for primes. Therefore the claim $\text{(*)}$ follows. ////

Elaborating this argument, we find that

$$a_k \equiv \frac{2^{5p_k}}{5p_k} \equiv \tilde{a}_k + \frac{32}{5p_k} \pmod{1}$$

where $\tilde{a}_k$ satisfies

$$\tilde{a}_k = \begin{cases} 0, & \text{if } p_k \equiv 1, 9, 13, 17 \pmod{20} \\ 1/5, & \text{if } p_k \equiv 11 \pmod{20} \\ 2/5, & \text{if } p_k \equiv 3 \pmod{20} \\ 3/5, & \text{if } p_k \equiv 7 \pmod{20} \\ 4/5, & \text{if } p_k \equiv 19 \pmod{20} \end{cases}.$$

Thus by the PNT for arithmetic progression again, we have the following convergence in distribution:

$$\frac{1}{n} \sum_{k=1}^{n} \delta_{a_k} \xrightarrow{d} \frac{1}{2}\delta_{0} + \frac{1}{8}\sum_{j=1}^{4} \delta_{j/5} \quad \text{as } n \to \infty$$

The following numerical simulation using first 1,000,000 terms of $(a_k)$ clearly demonstrates this behavior:

• Good job! I, too, noticed "horizontal dashed lines" at heights $k/5$, $k=0,1,2,3,4$ when I gave Mathematica the command to ListPlot values of Mod[2^n,n]/n. But I didn't get that those lines come from the case $n=5p$. – Jyrki Lahtonen Oct 14 '16 at 11:41
• @JyrkiLahtonen, Thank you! I chose the number $5$ out of no clear reason (except that the proof becomes simpler), but now your graph explains why this choice works. – Sangchul Lee Oct 14 '16 at 11:52
• @SangchulLee , First thank you for solving the problem. Second could you take a look at this related topic: math.stackexchange.com/questions/1965123/… I'm still not sure I understand your proof so I'll re-read it. I don't think I'll manage to prove the above on my own, but I'll try! – Aurelian Florea Oct 14 '16 at 13:39
• @SangchulLee, Isn't sum((2^n mod n)/n^2) suppose to equal sum((1/n^2)*{2^n/n}) rather than sum((1/n)*{2^n/n}) and then equal to sum(ak/25pk^2) – Aurelian Florea Oct 17 '16 at 9:19
• @SangchulLee Have you noticed and understood my previous comment? – Aurelian Florea Oct 18 '16 at 9:21

Not an answer, just for insight. Here is a plot of the $1000$ first $\dfrac{2^n\mod n}n$, to substantiate the "uniform distribution" hypothesis.

The second plot is the prefix sum, which is a straight line for a uniform distribution. The average of the values is $0.287$ (not very close to $0.5$).

Unfortunately, this empirical data isn't sufficiently conclusive.

Here are histograms of the fractional part of the $10000$, $100000$ and $1000000$ first values. Beware that the frequency axis is logarithmic. There is a strong bias towards the small values (one third below $0.01$), but this doesn't seem to worsen with $n$.

I conjecture that for any $n$, at least $50\%$ of the values of $\{2^k/k\}$ are above $0.1$.

Very interestingly, the histogram of the values $2^n\bmod n$ themselves shows very strong lines for all powers of $2$.

This leads us to the next histogram, where the moduli that are powers of $2$ have been ignored. There is no more need for a logarithmic scale and the distribution looks pretty flat now.

• Could you turn this into a histogram plot for the density in [0,1]? – H. H. Rugh Oct 12 '16 at 10:01
• @H.H.Rugh I added a cumulative plot instead. – Yves Daoust Oct 12 '16 at 10:09
• When you look at a plot of $\{2^x/x\}$ for real $x$ (fractional part), you get a lattice of quasi-parallel quasi-vertical quasi-line segments, the density of which goes increasing. Maybe one can establish something about the distribution of the intercepts with verticals $x=n$ ? – Yves Daoust Oct 14 '16 at 7:23
• Note that for every (odd) prime $n$ you have $2^n\equiv 2\bmod n$. That may be what's biasing the data toward the low side. – Gerry Myerson Oct 14 '16 at 8:49
• @GerryMyerson: amazing. These natural numbers never run out of surprises. – Yves Daoust Oct 14 '16 at 10:07

The point of this post is to give some plots that corroborate Sangchul Lee's argument. These were produced by Mathematica

The first plot lists the numbers $(2^n\mod n)/n$ for all multiples of $5$. You see the horizontal lines at multiples of $0.2$. They were already visible in the unrestricted plot, when I didn't restrict $n$ to multiples of five, but here they are easier to spot:

When we only include the values $n=5p$, the picture is even clearer:

Multiples of five are not the only structure in there. Restricting the choice of $n$ to numbers of the form $n=7p$ gives something quite similar.

• I had just started staring at the former image, when Sangchul Lee posted their solution. Look at that answer for an explanation. – Jyrki Lahtonen Oct 14 '16 at 11:51
• I hope that it is clear to all why the first image hides the pattern of the last. The first image was just vaguely hinted in the unrestricted plot but the last is already mostly lost in the noise. When you know what to look for, it is a lot easier to see it. – Jyrki Lahtonen Oct 14 '16 at 12:10
• If you liked this problem please check the more general case here: math.stackexchange.com/questions/1974148/… – Aurelian Florea Oct 18 '16 at 15:39


text:
## Calculus: Early Transcendentals 8th Edition

$\frac{16}{3}-4\sqrt{2}$
We begin with the integral in rectangular coordinates: $$\iint_{R}(2x-y)\,dA$$ The region in the first quadrant bounded by the circle $x^2+y^2=4$, the lines $x=0$ and $y=x$ have the bounds $0\leq r \leq2$ and $\pi/4\leq\theta\leq\pi/2$ so we can make the substitution into polar coordinates and add the bounds, getting: $$\int_{\pi/4}^{\pi/2}\bigg(\int_{0}^{2}(2r\cos\theta-r\sin\theta)\,r\,dr\bigg)d\theta$$ Cleaning this up and solving, we get: $$\int_{\pi/4}^{\pi/2}\bigg(\int_{0}^{2}r^2(2\cos\theta-\sin\theta)\,dr\bigg)d\theta\\=\int_{\pi/4}^{\pi/2}\bigg[\frac{r^3}{3}(2 \cos\theta-\sin\theta)\bigg]_{0}^{2}d\theta\\=\frac{8}{3}\int_{\pi/4}^{\pi/2}(2\cos\theta-\sin\theta)\,d\theta\\=\frac{8}{3}\bigg[2\sin\theta+\cos\theta\bigg]_{\pi/4}^{\pi/2}\\=\frac{16}{3}-4\sqrt{2}$$