...referred to as Cain’s notes), that I served as a primary text for an undergraduate level course in complex analysis. Throughout these notes I will make occasional references to results stated in these notes. The aim of my notes is to provide a few examples of applications of the residue theorem. The main goal is to illustrate how this theorem can be used to evaluate various types of integrals of real valued functions of real variable. Following Sec. 10.1 of Cain’s notes, let us recall that if C is a simple, closed contour and f is analytic within the region bounded by C except for finitely many points z0 , z1 , . . . , zk then k f (z)dz = 2πi C j=0 Resz=zj f (z), where Resz=a f (z) is the residue of f at a. 2 2.1 Evaluation of Real-Valued Integrals. Definite integrals involving trigonometric functions 2π We begin by briefly discussing integrals of the form F (sin at, cos bt)dt. 0 (1) Our method is easily adaptable for integrals over a different range, for example between 0 and π or between ±π. Given the form of an integrand in (1) one can reasonably hope that the integral results from the usual parameterization of the unit circle z = eit , 0 ≤ t ≤ 2π. So, let’s try z = eit . Then (see Sec. 3.3 of Cain’s notes), cos bt = z b + 1/z b eibt + e−ibt = , 2 2 sin at = eiat − e−iat z a − 1/z a = . 2i 2i Moreover, dz = ieit dt, so that dt = Putting all of this into (1) yields 2π dz . iz F (sin at, cos bt)dt...
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...nteThe Integral Approach® Courtesy of The Integral Institute® "Integral" means "inclusive, balanced, comprehensive." The Integral approach may be contrasted to other methods—mythic, rational-scientific, pluralistic—which, as they themselves announce, exclude other approaches as being inferior. They are thus, by definition, partial and incomplete. These latter methods, although widely accepted and dominant in the world's cultures, tend to generate partial analysis and incomplete solutions to problems. As such, they appear less efficient, less effective, and less balanced than the Integral approach. Like any truly fundamental advance, the Integral approach initially seems complicated but eventually is understood to be quite simple and even straightforward. It's like using a word processor: at first it is hard to learn, but eventually it becomes incredibly simple to use. The easiest way to understand the Integral approach is to remember that it was created by a cross-cultural comparison of most of the known forms of human inquiry. The result was a type of comprehensive map of human capacities. After this map was created (by looking at all the available research and evidence), it was discovered that this integral map had five major aspects to it. By learning to use these five major aspects, any thinker can fairly easily adopt a more comprehensive, effective, and integrally informed approach to specific problems and their solutions—from psychology to ecology, from business to politics...
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...Mathematics Syllabus Algebra: Algebra of complex numbers, addition, multiplication, conjugation, polar representation, properties of modulus and principal argument, triangle inequality, cube roots of unity, geometric interpretations. Quadratic equations with real coefficients, relations between roots and coefficients, formation of quadratic equations with given roots, symmetric functions of roots. Arithmetic, geometric and harmonic progressions, arithmetic, geometric and harmonic means, sums of finite arithmetic and geometric progressions, infinite geometric series, sums of squares and cubes of the first n natural numbers. Logarithms and their properties. Permutations and combinations, Binomial theorem for a positive integral index, properties of binomial coefficients. Matrices as a rectangular array of real numbers, equality of matrices, addition, multiplication by a scalar and product of matrices, transpose of a matrix, determinant of a square matrix of order up to three, inverse of a square matrix of order up to three, properties of these matrix operations, diagonal, symmetric and skew-symmetric matrices and their properties, solutions of simultaneous linear equations in two or three variables. Addition and multiplication rules of probability, conditional probability, Bayes Theorem, independence of events, computation of probability of events using permutations and combinations. Trigonometry: Trigonometric functions, their periodicity and graphs, addition...
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...Calculus From Wikipedia, the free encyclopedia This article is about the branch of mathematics. For other uses, see Calculus (disambiguation). Topics in Calculus Fundamental theorem Limits of functions Continuity Mean value theorem [show]Differential calculus [show]Integral calculus [show]Vector calculus [show]Multivariable calculus Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics focused on limits,functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modernmathematics education. It has two major branches,differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus is the study of change,[1] in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has widespread applications in science,economics, and engineering and can solve many problems for which algebra alone is insufficient. Historically, calculus was called "the calculus of infinitesimals", or "infinitesimal calculus". More generally, calculus (plural calculi) refers to any method or system of calculation guided by the symbolic manipulation of expressions. Some examples of other well-known calculi are propositional calculus...
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...section. (Discuss) Proposed since May 2011. | Topics in Calculus | Fundamental theorem Limits of functions Continuity Mean value theorem [show]Differential calculus | Derivative Change of variables Implicit differentiation Taylor's theorem Related rates Rules and identities:Power rule, Product rule, Quotient rule, Chain rule | [show]Integral calculus | IntegralLists of integrals Improper integrals Integration by: parts, disks, cylindrical shells, substitution, trigonometric substitution, partial fractions, changing order | [show]Vector calculus | Gradient Divergence Curl Laplacian Gradient theorem Green's theorem Stokes' theorem Divergence theorem | [show]Multivariable calculus | Matrix calculus Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian | | Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus is the study of change,[1] in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. A course in calculus is a gateway to other, more...
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...Calculus is the mathematical study of change,[1] in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. It has two major branches, differential calculus (concerning rates of change and slopes of curves), and integral calculus (concerning accumulation of quantities and the areas under and between curves); these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. Generally considered to have been founded in the 17th century by Isaac Newton and Gottfried Leibniz, today calculus has widespread uses in science, engineering and economics and can solve many problems that algebra alone cannot. Calculus is a part of modern mathematics education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has historically been called "the calculus of infinitesimals", or "infinitesimal calculus". The word "calculus" comes from Latin (calculus) and refers to a small stone used for counting. More generally, calculus (plural calculi) refers to any method or system of calculation guided by the symbolic manipulation of expressions. Some examples of other well-known calculi are propositional calculus, calculus of variations, lambda calculus, and...
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...3 Contour integrals and Cauchy’s Theorem 3.1 Line integrals of complex functions Our goal here will be to discuss integration of complex functions f (z) = u + iv, with particular regard to analytic functions. Of course, one way to think of integration is as antidifferentiation. But there is also the definite integral. For a function f (x) of a real variable x, we have the integral b f (x) dx. In case f (x) = u(x) + iv(x) is a complex-valued function of a a real variable x, the definite integral is the complex number obtained by b integrating the real and imaginary parts of f (x) separately, i.e. b b u(x) dx + i a f (x) dx = a v(x) dx. For vector fields F = (P, Q) in the plane we have a the line integral P dx+Q dy, where C is an oriented curve. In case P and C Q are complex-valued, in which case we call P dx + Q dy a complex 1-form, we again define the line integral by integrating the real and imaginary parts separately. Next we recall the basics of line integrals in the plane: 1. The vector field F = (P, Q) is a gradient vector field g, which we can write in terms of 1-forms as P dx + Q dy = dg, if and only if C P dx+Q dy only depends on the endpoints of C, equivalently if and only if C P dx+Q dy = 0 for every closed curve C. If P dx+Q dy = dg, and C has endpoints z0 and z1 , then we have the formula dg = g(z1 ) − g(z0 ). P dx + Q dy = C C 2. If D is a plane region with oriented boundary ∂D = C, then P dx + Q dy = C D ∂Q ∂P − ∂x ∂y dxdy. 3. If D is a simply connected...
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...antidifferentiation.For example, since the derivative of the distance travelled by a car is its velocity, the distance travelled is the antiderivative of the velocity of the car. * Types of integral: 1. Indefinite Integral: 2. Definite Integral: whereis the function, is antiderivative and is an arbitrary constant * Properties of Indefinite Integral Example Integrate the following function * The Integral of Trigonometric Functions Standard form | General form Where : | | | | | | | | | | | | | Remark: must be a LINEAR FUNCTION which means Example Evaluate the following: 6.2 METHOD OF INTEGRATION * Integration By Substitution * For product with the same type of functions * Steps: i) Substitute and to obtain the integral ii) Evaluate by integrating with respect to iii) Example Replace by in the resulting expression * Integration By Parts * For product with the different types of functions * Where are both functions of Example NOTE: 1. The choice must be such that the u part becomes a constant after successive differentiation 2. The dv part can be integrated from standard integrals 3. Normally, we give priority to the following functions as u part by following LIATE order: | Functions | Example | a) | Logarithmic function | | b) | Inverse trigonometric...
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...work, we present an extremely efficient approach for a fast numerical evaluation of highly oscillatory spherical Bessel integrals occurring in the analytic expressions of the so-called molecular multi-center integrals over exponential type functions. The approach is based on the Slevinsky-Safouhi formulae for higher derivatives applied to spherical Bessel functions and on extrapolation methods combined with practical properties of sine and cosine functions. Recurrence relations are used for computing the approximations of the spherical Bessel integrals, allowing for a control of accuracy and the stability of the algorithm. The computer algebra system Maple was used in our development, mainly to prove the applicability of the extrapolation methods. Among molecular multi-center integrals, the three-center nuclear attraction and four-center two-electron Coulomb and exchange integrals are undoubtedly the most difficult ones to evaluate rapidly to a high pre-determined accuracy. These integrals are required for both density functional and ab initio calculations. Already for small molecules, many millions of them have to be computed. As the molecular system gets larger, the computation of these integrals becomes one of the most laborious and time consuming steps in molecular electronic structure calculation. Improvement of the computational methods of molecular integrals would be indispensable to a further development in computational studies of large molecular systems. Convergence properties...
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...BCO 116 Elementary Calculus 2Ch, 3 ECTS Academic Year 2015-2016 Course Description This course is an introduction to calculus and is intended to familiarize students with the mathematical thinking. It focuses on topics of calculus that are relevant to students in the managerial and business sciences. The main goals of the course are to develop technical skills allowing the analysis of real life problems, to provide the tools necessary to formulate, analyse and implement a simple quantitative model to support a business decision, and to understand articles and books on business analysis Starting with preliminaries to calculus, and limits, we follow with derivatives and its applications to real life problems, and integration. This course covers also functions with more than one variable, differential equations, and optimization. Basic requirement. Students must have a good background on algebra and arithmetic, as well as a good understanding of mathematical functions and their applications to practical problems. Course Objectives * To builds skills and proficiency in methods of calculus * To understand concepts, formulas and techniques of calculus through exercises and applied examples * To be able to translate real-world problems to mathematical language and models * To acquire ease in identifying the different kind of problem and the appropriated rule to solve it * To interpret results of calculus * To apply analytical methods of calculus...
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...a machining process improvement project that I participated at XXXXXl. I was working as a Manufacturing Process Engineer with machining process, and in this particular case I made a validation of new cutting fluid for shaving process. Background (2) The main purpose of this project was validate a new kind of cutting fluid for shaving process of the gears for vehicle transmissions at Eaton (http://www.). These gears are the main parts of the transmission, which is manufactured in many machining steps, thermal treatment, assembly, and the project aim was improvement / cost reduction of cutting fluid usage. (3) Due to approval problems in the operation of shaving, the manufacturing team exchanged the current oil that time (integral oil plus chlorinated paraffin – usually used in shaving process) for new oil (soluble oil – called FLUCOR 405e) with thirty percent of concentration and seventy percent of water. There was no follow up for all costs involved and performance in the operation that could ensure that the usage of the new oil was advantageous for the company. (4) The facilitators were EHS department (Environment Health and Safety), Lubrication department, and Manufacturing department. As internal clients two areas of the company were considered the main departments with possible benefits, Manufacturing and Management tools. (5) Manager EHS Department Quality Engineerig Process Engineering Accounting / Cost Environment Safety Roughness Approval ...
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... . . . . . . . . . . . . . . . . . . . iii 1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Lines and Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Vector Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Space Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Multivariable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Directional Derivatives and Gradients . . . . . . . . . . . . . . . . . . . . . . . 77 Tangent Planes and Normal Vectors . . . . . . . . . . . . . . . . . . . . . . . 81 Extremal Values of Multivariable Functions . . . . . . . . . . . . . . . . . . . 91 Lagrange Multipliers* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Double Integrals over Rectangles . . . . . . . . . . . . . . . . . . . . . . . . 105 Double Integrals over General Regions . . . . . . . . . . . . . . . . . . . . . 111 Double Integrals in Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . 121 Triple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Triple Integrals in Cylindrical and Spherical Coordinates . . . . . . ....
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...TLFeBOOK WHAT READERS ARE SAYIN6 "I wish I had had this book when I needed it most, which was during my pre-med classes. I t could have also been a great tool for me in a few medical school courses." Or. Kellie Aosley8 Recent Hedical school &a&ate "CALCULUS FOR THE UTTERLY CONFUSED has proven to be a wonderful review enabling me t o move forward in application of calculus and advanced topics in mathematics. I found it easy t o use and great as a reference for those darker aspects of calculus. I' Aaron Ladeville, Ekyiheeriky Student 'I1am so thankful for CALCULUS FOR THE UTTERLY CONFUSED! I started out Clueless but ended with an All' Erika Dickstein8 0usihess school Student "As a non-traditional student one thing I have learned is the Especially in importance of material supplementary t o texts. calculus it helps to have a second source, especially one as lucid and fun t o read as CALCULUS FOR THE UTTERtY CONFUSED. Anyone, whether you are a math weenie or not, will get something out of this book. With this book, your chances of survival in the calculus jungle are greatly increased.'I Brad &3~ker, Physics Student Other books i the Utterly Conhrsed Series include: n Financial Planning for the Utterly Confrcsed, Fifth Edition Job Hunting for the Utterly Confrcred Physics for the Utterly Confrred CALCULUS FOR THE UTTERLY CONFUSED Robert M. Oman Daniel M. Oman McGraw-Hill New York San Francisco Washington, D.C. Auckland Bogoth Caracas Lisbon...
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...Probability, Mean and Median In the last section, we considered (probability) density functions. We went on to discuss their relationship with cumulative distribution functions. The goal of this section is to take a closer look at densities, introduce some common distributions and discuss the mean and median. Recall, we define probabilities as follows: Proportion of population for Area under the graph of p ( x ) between a and b which x is between a and b p( x)dx a b The cumulative distribution function gives the proportion of the population that has values below t. That is, t P (t ) p( x)dx Proportion of population having values of x below t When answering some questions involving probabilities, both the density function and the cumulative distribution can be used, as the next example illustrates. Example 1: Consider the graph of the function p(x). p x 0.2 0.1 2 4 6 8 10 x Figure 1: The graph of the function p(x) a. Explain why the function is a probability density function. b. Use the graph to find P(X < 3) c. Use the graph to find P(3 § X § 8) 1 Solution: a. Recall, a function is a probability density function if the area under the curve is equal to 1 and all of the values of p(x) are non-negative. It is immediately clear that the values of p(x) are non-negative. To verify that the area under the curve is equal to 1, we recognize that the graph above can be viewed as a triangle. Its...
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...Taylor’s theorem (proof), Remainder terms, Asymptote & Curvature, Partial derivatives, Chain rule, Differentiation of Implicit functions, Exact differentials, Tangents and Normals, Maxima, Approximations, Differentiation under integral sign, Jacobians and transformations of coordinates. Module II: Integral Calculus Fundamental theorems, Reduction formulae, Properties of definite integrals, Applications to length, area, volume, surface of revolution, improper integrals, Multiple Integrals-Double integrals, Applications to areas, volumes. Module III: Ordinary Differential Equations Formation of ODEs, Definition of order, degree & solutions, ODE of first order : Method of separation of variables, homogeneous and non homogeneous equations, Exactness & integrating factors, Linear equations & Bernoulli equations, General linear ODE of nth order, Solution of homogeneous equations, Operator method, Method of undetermined coefficients, Solution of simple simultaneous ODE. Module IV: Vector Calculus Scalar and Vector Field, Derivative of a Vector, Gradient, Directional Derivative, Divergence and Curl and their Physical Significance, Arc Length, Tangent, Directional Derivative, Evaluation of Line Integral, Green’s Theorem in...
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