The Infinitesimal M. Noether Theorem for Singularities
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0.999… = 1 an Infinitesimal Explanation Bryan Dawson
0 1 2 0.9999999999999999 0.999… = 1 An Infinitesimal Explanation Bryan Dawson know the proofs, but I still don’t What exactly does that mean? Just as real num- believe it.” Those words were uttered bers have decimal expansions, with one digit for each to me by a very good undergraduate integer power of 10, so do hyperreal numbers. But the mathematics major regarding hyperreals contain “infinite integers,” so there are digits This fact is possibly the most-argued- representing not just (the 237th digit past “Iabout result of arithmetic, one that can evoke great the decimal point) and (the 12,598th digit), passion. But why? but also (the Yth digit past the decimal point), According to Robert Ely [2] (see also Tall and where is a negative infinite hyperreal integer. Vinner [4]), the answer for some students lies in their We have four 0s followed by a 1 in intuition about the infinitely small: While they may the fifth decimal place, and also where understand that the difference between and 1 is represents zeros, followed by a 1 in the Yth less than any positive real number, they still perceive a decimal place. (Since we’ll see later that not all infinite nonzero but infinitely small difference—an infinitesimal hyperreal integers are equal, a more precise, but also difference—between the two. And it’s not just uglier, notation would be students; most professional mathematicians have not or formally studied infinitesimals and their larger setting, the hyperreal numbers, and as a result sometimes Confused? Perhaps a little background information wonder . -
Notes on Euler's Work on Divergent Factorial Series and Their Associated
Indian J. Pure Appl. Math., 41(1): 39-66, February 2010 °c Indian National Science Academy NOTES ON EULER’S WORK ON DIVERGENT FACTORIAL SERIES AND THEIR ASSOCIATED CONTINUED FRACTIONS Trond Digernes¤ and V. S. Varadarajan¤¤ ¤University of Trondheim, Trondheim, Norway e-mail: [email protected] ¤¤University of California, Los Angeles, CA, USA e-mail: [email protected] Abstract Factorial series which diverge everywhere were first considered by Euler from the point of view of summing divergent series. He discovered a way to sum such series and was led to certain integrals and continued fractions. His method of summation was essentialy what we call Borel summation now. In this paper, we discuss these aspects of Euler’s work from the modern perspective. Key words Divergent series, factorial series, continued fractions, hypergeometric continued fractions, Sturmian sequences. 1. Introductory Remarks Euler was the first mathematician to develop a systematic theory of divergent se- ries. In his great 1760 paper De seriebus divergentibus [1, 2] and in his letters to Bernoulli he championed the view, which was truly revolutionary for his epoch, that one should be able to assign a numerical value to any divergent series, thus allowing the possibility of working systematically with them (see [3]). He antic- ipated by over a century the methods of summation of divergent series which are known today as the summation methods of Cesaro, Holder,¨ Abel, Euler, Borel, and so on. Eventually his views would find their proper place in the modern theory of divergent series [4]. But from the beginning Euler realized that almost none of his methods could be applied to the series X1 1 ¡ 1!x + 2!x2 ¡ 3!x3 + ::: = (¡1)nn!xn (1) n=0 40 TROND DIGERNES AND V. -
Understanding Student Use of Differentials in Physics Integration Problems
PHYSICAL REVIEW SPECIAL TOPICS - PHYSICS EDUCATION RESEARCH 9, 020108 (2013) Understanding student use of differentials in physics integration problems Dehui Hu and N. Sanjay Rebello* Department of Physics, 116 Cardwell Hall, Kansas State University, Manhattan, Kansas 66506-2601, USA (Received 22 January 2013; published 26 July 2013) This study focuses on students’ use of the mathematical concept of differentials in physics problem solving. For instance, in electrostatics, students need to set up an integral to find the electric field due to a charged bar, an activity that involves the application of mathematical differentials (e.g., dr, dq). In this paper we aim to explore students’ reasoning about the differential concept in physics problems. We conducted group teaching or learning interviews with 13 engineering students enrolled in a second- semester calculus-based physics course. We amalgamated two frameworks—the resources framework and the conceptual metaphor framework—to analyze students’ reasoning about differential concept. Categorizing the mathematical resources involved in students’ mathematical thinking in physics provides us deeper insights into how students use mathematics in physics. Identifying the conceptual metaphors in students’ discourse illustrates the role of concrete experiential notions in students’ construction of mathematical reasoning. These two frameworks serve different purposes, and we illustrate how they can be pieced together to provide a better understanding of students’ mathematical thinking in physics. DOI: 10.1103/PhysRevSTPER.9.020108 PACS numbers: 01.40.Àd mathematical symbols [3]. For instance, mathematicians I. INTRODUCTION often use x; y asR variables and the integrals are often written Mathematical integration is widely used in many intro- in the form fðxÞdx, whereas in physics the variables ductory and upper-division physics courses. -
Leonhard Euler: His Life, the Man, and His Works∗
SIAM REVIEW c 2008 Walter Gautschi Vol. 50, No. 1, pp. 3–33 Leonhard Euler: His Life, the Man, and His Works∗ Walter Gautschi† Abstract. On the occasion of the 300th anniversary (on April 15, 2007) of Euler’s birth, an attempt is made to bring Euler’s genius to the attention of a broad segment of the educated public. The three stations of his life—Basel, St. Petersburg, andBerlin—are sketchedandthe principal works identified in more or less chronological order. To convey a flavor of his work andits impact on modernscience, a few of Euler’s memorable contributions are selected anddiscussedinmore detail. Remarks on Euler’s personality, intellect, andcraftsmanship roundout the presentation. Key words. LeonhardEuler, sketch of Euler’s life, works, andpersonality AMS subject classification. 01A50 DOI. 10.1137/070702710 Seh ich die Werke der Meister an, So sehe ich, was sie getan; Betracht ich meine Siebensachen, Seh ich, was ich h¨att sollen machen. –Goethe, Weimar 1814/1815 1. Introduction. It is a virtually impossible task to do justice, in a short span of time and space, to the great genius of Leonhard Euler. All we can do, in this lecture, is to bring across some glimpses of Euler’s incredibly voluminous and diverse work, which today fills 74 massive volumes of the Opera omnia (with two more to come). Nine additional volumes of correspondence are planned and have already appeared in part, and about seven volumes of notebooks and diaries still await editing! We begin in section 2 with a brief outline of Euler’s life, going through the three stations of his life: Basel, St. -
Calculus Terminology
AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential -
Euler and His Work on Infinite Series
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 44, Number 4, October 2007, Pages 515–539 S 0273-0979(07)01175-5 Article electronically published on June 26, 2007 EULER AND HIS WORK ON INFINITE SERIES V. S. VARADARAJAN For the 300th anniversary of Leonhard Euler’s birth Table of contents 1. Introduction 2. Zeta values 3. Divergent series 4. Summation formula 5. Concluding remarks 1. Introduction Leonhard Euler is one of the greatest and most astounding icons in the history of science. His work, dating back to the early eighteenth century, is still with us, very much alive and generating intense interest. Like Shakespeare and Mozart, he has remained fresh and captivating because of his personality as well as his ideas and achievements in mathematics. The reasons for this phenomenon lie in his universality, his uniqueness, and the immense output he left behind in papers, correspondence, diaries, and other memorabilia. Opera Omnia [E], his collected works and correspondence, is still in the process of completion, close to eighty volumes and 31,000+ pages and counting. A volume of brief summaries of his letters runs to several hundred pages. It is hard to comprehend the prodigious energy and creativity of this man who fueled such a monumental output. Even more remarkable, and in stark contrast to men like Newton and Gauss, is the sunny and equable temperament that informed all of his work, his correspondence, and his interactions with other people, both common and scientific. It was often said of him that he did mathematics as other people breathed, effortlessly and continuously. -
Infinitesimal Calculus
Infinitesimal Calculus Δy ΔxΔy and “cannot stand” Δx • Derivative of the sum/difference of two functions (x + Δx) ± (y + Δy) = (x + y) + Δx + Δy ∴ we have a change of Δx + Δy. • Derivative of the product of two functions (x + Δx)(y + Δy) = xy + Δxy + xΔy + ΔxΔy ∴ we have a change of Δxy + xΔy. • Derivative of the product of three functions (x + Δx)(y + Δy)(z + Δz) = xyz + Δxyz + xΔyz + xyΔz + xΔyΔz + ΔxΔyz + xΔyΔz + ΔxΔyΔ ∴ we have a change of Δxyz + xΔyz + xyΔz. • Derivative of the quotient of three functions x Let u = . Then by the product rule above, yu = x yields y uΔy + yΔu = Δx. Substituting for u its value, we have xΔy Δxy − xΔy + yΔu = Δx. Finding the value of Δu , we have y y2 • Derivative of a power function (and the “chain rule”) Let y = x m . ∴ y = x ⋅ x ⋅ x ⋅...⋅ x (m times). By a generalization of the product rule, Δy = (xm−1Δx)(x m−1Δx)(x m−1Δx)...⋅ (xm −1Δx) m times. ∴ we have Δy = mx m−1Δx. • Derivative of the logarithmic function Let y = xn , n being constant. Then log y = nlog x. Differentiating y = xn , we have dy dy dy y y dy = nxn−1dx, or n = = = , since xn−1 = . Again, whatever n−1 y dx x dx dx x x x the differentials of log x and log y are, we have d(log y) = n ⋅ d(log x), or d(log y) n = . Placing these values of n equal to each other, we obtain d(log x) dy d(log y) y dy = . -
On the Euler Integral for the Positive and Negative Factorial
On the Euler Integral for the positive and negative Factorial Tai-Choon Yoon ∗ and Yina Yoon (Dated: Dec. 13th., 2020) Abstract We reviewed the Euler integral for the factorial, Gauss’ Pi function, Legendre’s gamma function and beta function, and found that gamma function is defective in Γ(0) and Γ( x) − because they are undefined or indefinable. And we came to a conclusion that the definition of a negative factorial, that covers the domain of the negative space, is needed to the Euler integral for the factorial, as well as the Euler Y function and the Euler Z function, that supersede Legendre’s gamma function and beta function. (Subject Class: 05A10, 11S80) A. The positive factorial and the Euler Y function Leonhard Euler (1707–1783) developed a transcendental progression in 1730[1] 1, which is read xedx (1 x)n. (1) Z − f From this, Euler transformed the above by changing e to g for generalization into f x g dx (1 x)n. (2) Z − Whence, Euler set f = 1 and g = 0, and got an integral for the factorial (!) 2, dx ( lx )n, (3) Z − where l represents logarithm . This is called the Euler integral of the second kind 3, and the equation (1) is called the Euler integral of the first kind. 4, 5 Rewriting the formula (3) as follows with limitation of domain for a positive half space, 1 1 n ln dx, n 0. (4) Z0 x ≥ ∗ Electronic address: [email protected] 1 “On Transcendental progressions that is, those whose general terms cannot be given algebraically” by Leonhard Euler p.3 2 ibid. -
Leonhard Euler - Wikipedia, the Free Encyclopedia Page 1 of 14
Leonhard Euler - Wikipedia, the free encyclopedia Page 1 of 14 Leonhard Euler From Wikipedia, the free encyclopedia Leonhard Euler ( German pronunciation: [l]; English Leonhard Euler approximation, "Oiler" [1] 15 April 1707 – 18 September 1783) was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function.[2] He is also renowned for his work in mechanics, fluid dynamics, optics, and astronomy. Euler spent most of his adult life in St. Petersburg, Russia, and in Berlin, Prussia. He is considered to be the preeminent mathematician of the 18th century, and one of the greatest of all time. He is also one of the most prolific mathematicians ever; his collected works fill 60–80 quarto volumes. [3] A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is our teacher in all things," which has also been translated as "Read Portrait by Emanuel Handmann 1756(?) Euler, read Euler, he is the master of us all." [4] Born 15 April 1707 Euler was featured on the sixth series of the Swiss 10- Basel, Switzerland franc banknote and on numerous Swiss, German, and Died Russian postage stamps. The asteroid 2002 Euler was 18 September 1783 (aged 76) named in his honor. He is also commemorated by the [OS: 7 September 1783] Lutheran Church on their Calendar of Saints on 24 St. Petersburg, Russia May – he was a devout Christian (and believer in Residence Prussia, Russia biblical inerrancy) who wrote apologetics and argued Switzerland [5] forcefully against the prominent atheists of his time. -
Arxiv:1404.5658V1 [Math.LO] 22 Apr 2014 03F55
TOWARD A CLARITY OF THE EXTREME VALUE THEOREM KARIN U. KATZ, MIKHAIL G. KATZ, AND TARAS KUDRYK Abstract. We apply a framework developed by C. S. Peirce to analyze the concept of clarity, so as to examine a pair of rival mathematical approaches to a typical result in analysis. Namely, we compare an intuitionist and an infinitesimal approaches to the extreme value theorem. We argue that a given pre-mathematical phenomenon may have several aspects that are not necessarily captured by a single formalisation, pointing to a complementar- ity rather than a rivalry of the approaches. Contents 1. Introduction 2 1.1. A historical re-appraisal 2 1.2. Practice and ontology 3 2. Grades of clarity according to Peirce 4 3. Perceptual continuity 5 4. Constructive clarity 7 5. Counterexample to the existence of a maximum 9 6. Reuniting the antipodes 10 7. Kronecker and constructivism 11 8. Infinitesimal clarity 13 8.1. Nominalistic reconstructions 13 8.2. Klein on rivalry of continua 14 arXiv:1404.5658v1 [math.LO] 22 Apr 2014 8.3. Formalizing Leibniz 15 8.4. Ultrapower 16 9. Hyperreal extreme value theorem 16 10. Approaches and invitations 18 11. Conclusion 19 References 19 2000 Mathematics Subject Classification. Primary 26E35; 00A30, 01A85, 03F55. Key words and phrases. Benacerraf, Bishop, Cauchy, constructive analysis, con- tinuity, extreme value theorem, grades of clarity, hyperreal, infinitesimal, Kaestner, Kronecker, law of excluded middle, ontology, Peirce, principle of unique choice, procedure, trichotomy, uniqueness paradigm. 1 2 KARINU. KATZ, MIKHAILG. KATZ, ANDTARASKUDRYK 1. Introduction German physicist G. Lichtenberg (1742 - 1799) was born less than a decade after the publication of the cleric George Berkeley’s tract The Analyst, and would have certainly been influenced by it or at least aware of it. -
Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World Reviewed by Slava Gerovitch
BOOK REVIEW Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World Reviewed by Slava Gerovitch Infinitesimal: How a Dangerous Mathematical Theory ing, from the chaos of imprecise analogies to the order Shaped the Modern World of disciplined reasoning. Yet, as Amir Alexander argues Amir Alexander in his fascinating book Infinitesimal: How a Dangerous Scientific American/Farrar, Straus and Giroux, April 2014 Mathematical Theory Shaped the Modern World, it was US$27.00/US$16.00, 368 pages precisely the champions of offensive infinitesimals who Hardcover ISBN-13: 978-0-37-417681-5 propelled mathematics forward, while the rational critics Paperback ISBN-13: 978-0-37-453499-8 slowed the development of mathematical thought. More- over, the debate over infinitesimals reflected a larger clash From today’s perspective, a mathematical technique that in European culture between religious dogma and intellec- lacks rigor or leads to paradoxes is a contradiction in tual pluralism and between the proponents of traditional terms. It must be expelled from mathematics, lest it dis- order and the defenders of new liberties. credit the profession, sow chaos, and put a large stain on Known since antiquity, the concept of indivisibles gave the shining surface of eternal truth. This is precisely what rise to Zeno’s paradoxes, including the famous “Achilles the leading mathematicians of the most learned Catholic and the Tortoise” conundrum, and was subjected to scath- order, the Jesuits, said about the “method of indivisibles”, ing philosophical critique by Plato and Aristotle. Archime- a dubious procedure of calculating areas and volumes by des used the method of indivisibles with considerable suc- representing plane figures or solids as a composition of in- cess, but even he, once a desired volume was calculated, divisible lines or planes, “infinitesimals”. -
List of Mathematical Symbols by Subject from Wikipedia, the Free Encyclopedia
List of mathematical symbols by subject From Wikipedia, the free encyclopedia This list of mathematical symbols by subject shows a selection of the most common symbols that are used in modern mathematical notation within formulas, grouped by mathematical topic. As it is virtually impossible to list all the symbols ever used in mathematics, only those symbols which occur often in mathematics or mathematics education are included. Many of the characters are standardized, for example in DIN 1302 General mathematical symbols or DIN EN ISO 80000-2 Quantities and units – Part 2: Mathematical signs for science and technology. The following list is largely limited to non-alphanumeric characters. It is divided by areas of mathematics and grouped within sub-regions. Some symbols have a different meaning depending on the context and appear accordingly several times in the list. Further information on the symbols and their meaning can be found in the respective linked articles. Contents 1 Guide 2 Set theory 2.1 Definition symbols 2.2 Set construction 2.3 Set operations 2.4 Set relations 2.5 Number sets 2.6 Cardinality 3 Arithmetic 3.1 Arithmetic operators 3.2 Equality signs 3.3 Comparison 3.4 Divisibility 3.5 Intervals 3.6 Elementary functions 3.7 Complex numbers 3.8 Mathematical constants 4 Calculus 4.1 Sequences and series 4.2 Functions 4.3 Limits 4.4 Asymptotic behaviour 4.5 Differential calculus 4.6 Integral calculus 4.7 Vector calculus 4.8 Topology 4.9 Functional analysis 5 Linear algebra and geometry 5.1 Elementary geometry 5.2 Vectors and matrices 5.3 Vector calculus 5.4 Matrix calculus 5.5 Vector spaces 6 Algebra 6.1 Relations 6.2 Group theory 6.3 Field theory 6.4 Ring theory 7 Combinatorics 8 Stochastics 8.1 Probability theory 8.2 Statistics 9 Logic 9.1 Operators 9.2 Quantifiers 9.3 Deduction symbols 10 See also 11 References 12 External links Guide The following information is provided for each mathematical symbol: Symbol: The symbol as it is represented by LaTeX.