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Dave's short course in trigonometry Table of Contents Who should take this course? Trigonometry for you Your background How to learn trigonometry Applications of trigonometry Astronomy and geography Engineering and physics Mathematics and its applications What is trigonometry? Trigonometry as computational geometry Angle measurement and tables Background on geometry The Pythagorean theorem An explanation of the Pythagorean theorem Similar triangles Angle measurement The concept of angle Radians and arc length Exercises, hints, and answers About digits of accuracy Chords What is a chord? Ptolemy’s sum and difference formulas Ptolemy’s theorem The sum formula for sines The other sum and difference formulas Summary of trigonometric formulas Formulas for arcs and sectors of circles Formulas for right triangles Formulas for oblique triangles Formulas for areas of triangles Summary of trigonometric identities More important identities Less important identities Truly obscure identities About the Java applet.

Mathematics Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day. History Evolution Etymology Definitions of mathematics

Physics Various examples of physical phenomena Physics is one of the oldest academic disciplines, perhaps the oldest through its inclusion of astronomy. Over the last two millennia, physics was a part of natural philosophy along with chemistry, certain branches of mathematics, and biology, but during the Scientific Revolution in the 17th century, the natural sciences emerged as unique research programs in their own right.[b] Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms of other sciences while opening new avenues of research in areas such as mathematics and philosophy. Physics also makes significant contributions through advances in new technologies that arise from theoretical breakthroughs. History Ancient astronomy Astronomy is the oldest of the natural sciences. Natural philosophy Classical physics Modern physics

Math 6620: Perturbation Methods Welcome to the home page for Perturbation Methods, Spring 2006. This site will be used to provide homework assignments, supplementary notes, announcements, and other useful information. Please check regularly for updates. Last Update: 4-24-06 Announcements and Recent Postings Supplemental Notes on Laplace's Method.Here are some problems which I had intended to put on the final exam. Lecture Notes Contact Information Chris Wahle Office: Amos Eaton 308 Office Hours: Monday and Thursday, 2:00-3:00pm, or by appointment e-mail: wahlec@rpi.edu Link to homepage Course Information Homework homework 1 . Supplementary Notes and Examples Mathematical Atlas: A gateway to Mathematics Welcome! This is a collection of short articles designed to provide an introduction to the areas of modern mathematics and pointers to further information, as well as answers to some common (or not!) questions. The material is arranged in a hierarchy of disciplines, each with its own index page ("blue pages"). For resources useful in all areas of mathematics try 00: General Mathematics. There is a backlog of articles awaiting editing before they are referenced in the blue pages, but you are welcome to snoop around VIRUS WARNING: The Mathematical Atlas receives but does not send mail using the math-atlas.org domain name. Please bookmark any pages at this site with the URL This URL forces frames; for a frame-free version use

Non-Euclidean Geometry In three dimensions, there are three classes of constant curvature geometries. All are based on the first four of Euclid's postulates, but each uses its own version of the parallel postulate. The "flat" geometry of everyday intuition is called Euclidean geometry (or parabolic geometry), and the non-Euclidean geometries are called hyperbolic geometry (or Lobachevsky-Bolyai-Gauss geometry) and elliptic geometry (or Riemannian geometry). List of theorems This is a list of theorems, by Wikipedia page. See also Most of the results below come from pure mathematics, but some are from theoretical physics, economics, and other applied fields. 0–9 A B C D E F G H I J K L M N O P Q R S T U V W Z

MATH 450 Home Page MATH 450 Applied Mathematics 1 MWF 10-10:50AM in Wilson Hall 1-124 Announcements For a general overview of the course, download the Course Policies . The FINAL EXAM is 8:00-9:50AM on Monday, December 10. Supplementary Texts: Elementary Differential Equations texts (pick your favorite). Students are NOT required to buy the supplementary texts. Imaging maths - Inside the Klein bottle September 2003 My first practical experience with a Klein bottle was a failure. When I was still at school I came across a picture of a Klein bottle in a weekly journal. What to do with such a bottle? To my delight, this primitive but interesting thermometer did indeed show signs of life during the day. The Klein bottle was discovered in 1882 by Felix Klein  and since then has joined the gallery of popular mathematical shapes known to the general public outside the "ivory tower". From the Möbius band to the projective plane The Möbius band is the simplest one-sided surface, and is easy to make out of a strip of paper. The band was discovered in 1858 by the German astronomer and mathematician August Ferdinand Möbius. Topology is the mathematical discipline which investigates those properties of shapes which do not change under continuous bending and stretching. Despite its simplicity the Möbius band was a genuine mathematical discovery. The Klein bottle is not a doughnut Felix Klein F.

Non-Euclidean geometry Behavior of lines with a common perpendicular in each of the three types of geometry In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises when either the metric requirement is relaxed, or the parallel postulate is set aside. In the latter case one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras which give rise to kinematic geometries that have also been called non-Euclidean geometry. Another way to describe the differences between these geometries is to consider two straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to a third line: History Early history Terminology

Theorem Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from the hypotheses. In light of the interpretation of proof as justification of truth, the conclusion is often viewed as a necessary consequence of the hypotheses, namely, that the conclusion is true in case the hypotheses are true, without any further assumptions. However, the conditional could be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol. Although they can be written in a completely symbolic form, for example, within the propositional calculus, theorems are often expressed in a natural language such as English. Informal account of theorems Logically, many theorems are of the form of an indicative conditional: if A, then B. To be proven, a theorem must be expressible as a precise, formal statement. Other theorems have a known proof that cannot easily be written down. Layout

Math 6730 Math 6730 - Asymptotic and Perturbation Methods Time: 2-3:20pm T,TH Place: JWB 222 Course Description: In this course, we will discuss the 4 basic problems of singular perturbation theory, namely singular boundary value problems, singular initial value problems, multiple time scale problems, and multiple space scale problems. Text: M. Other References: J. Class Schedule: Because I have a lot of travel this semester, we will meet for makeup classes at 3:30-4:50 on Sept. 19, Oct. 3, 24, Nov. 7, 14, 19, and Dec. 2 in LCB 323. Notes: Maple code for regular perturbation of gravitational trajectory Maple code for higher order matching Maple code for problem with multiple boundary layers Notes on corner layer analysis Notes on quasi-steady state analysis Maple code for van der pol multiscaling analysis Maple code for forced Duffing equation multiscaling analysis Notes on averaging the pendulum equation For more information contact J. E-mail: keener@math.utah.edu

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