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 موضوع: كتاب Mathematical Formulas for Industrial and Mechanical Engineering الثلاثاء 11 يناير 2022, 2:22 am  
 أخواني في الله أحضرت لكم كتاب Mathematical Formulas for Industrial and Mechanical Engineering Seifedine Kadry American University of the Middle East, Kuwait
و المحتوى كما يلي :
1 Symbols and Special Numbers In this chapter, several symbols used in mathematics are defined. Some special numbers are given with examples and many conversion formulas are studied. This chapter is essential to understand the next chapters. Topics discussed in this chapter are as follows: ● Basic mathematical symbols ● Base algebra symbols ● Linear algebra symbols ● Probability and statistics symbols ● Geometry symbols ● Set theory symbols ● Logic symbols ● Calculus symbols ● Numeral symbols ● Greek alphabet letters ● Roman numerals ● Special numbers like prime numbers ● Conversion formulas ● Basic area, perimeter, and volume formulas. 2 Elementary Algebra Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas arithmetic deals with specified numbers, algebra introduces quantities without fixed values known as variables. This use of variables entails a use of algebraic notation and an understanding of the general rules of the operators introduced in arithmetic. Unlike abstract algebra, elementary algebra is not concerned with algebraic structures outside the realm of real and complex numbers. The use of variables to denote quantities allows general relationships between quantities to be formally and concisely expressed, and thus enables solving a broader scope of problems. Most quantitative results in science and mathematics are expressed as algebraic equations. ● Sets of numbers ● Absolute value ● Basic properties of real numbers ● Logarithm ● Factorials ● Solving algebraic equations ● Intervals ● Complex numbers ● Euler’s formula 3 Linear Algebra Linear algebra is the branch of mathematics concerning vector spaces, often finite or countable infinite dimensional, as well as linear mappings between such spaces. Such an investigation is initially motivated by a system of linear equations in several unknowns. Such equations are naturally represented using the formalism of matrices and vectors. Linear algebra is central to both pure and applied mathematics. For instance, abstract algebra arises by relaxing the axioms of a vector space, leading to a number of generalizations. Functional analysis studies the infinitedimensional version of the theory of vector spaces. Combined with calculus, linear algebra facilitates the solution of linear systems of differential equations. Techniques from linear algebra are also used in analytic geometry, engineering, physics, natural sciences, computer science, computer animation, and the social sciences (particularly in economics). Because linear algebra is such a welldeveloped theory, nonlinear mathematical models are sometimes approximated by linear ones. Topics discussed in this chapter are as follows: ● Basic types of matrices ● Basic operations on matrices ● Determinants ● Sarrus rule ● Minors and cofactors ● Inverse matrix ● System of linear equations ● Cramer’s rule. 4 Analytic Geometry and Trigonometry Geometry is divided into two branches: analytic geometry and trigonometry. Trigonometry began as the computational component of geometry. For instance, one statement of plane geometry states that a triangle is determined by a side and two angles. In other words, given one side of a triangle and two angles in the triangle, then the other two sides and the remaining angle are determined. Trigonometry includes the methods for computing those other two sides. The remaining angle is easy to find since the sum of the three angles equals 180 degrees (usually written as 180). Analytic geometry is a branch of algebra that is used to model geometric objects—points, (straight) lines, and circles being the most basic of these. In plane analytic geometry (twodimensional), points are defined as ordered pairs of numbers, say, (x, y), while the straight lines are in turn defined as the sets of points that satisfy linear equations. Topics discussed in this chapter are as follows: ● Plane figures ● Solid figures ● Triangles ● Degrees or radians ● Table of natural trigonometric functions ● Trigonometry identities ● The inverse trigonometric functions ● Solutions of trigonometric equations ● Analytic geometry (in the plane, i.e., 2D) ● Vector 5 Calculus Calculus is the mathematical study of change, 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 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 welldefined limit. Calculus has widespread uses in science, economics, and engineering and can solve many problems that algebra alone cannot. Topics discussed in this chapter are as follows: ● Functions and their graphs ● Limits of functions ● Definition and properties of the derivative ● Table of derivatives ● Applications of derivative ● Indefinite integral ● Integrals of rational function ● Integrals of irrational function ● Integrals of trigonometric functions ● Integrals of hyperbolic functions ● Integrals of exponential and logarithmic functions ● Reduction formulas using integration by part ● Definite integral ● Improper integral ● Continuity of a function ● Partial fractions ● Properties of trigonometric functions ● Sequences and series ● Convergence tests for series ● Taylor and Maclaurin series ● Continuous Fourier series ● Double integrals ● Triple integrals ● Firstorder differential equation ● Secondorder differential equation ● Laplace transform ● Table of Laplace transforms 6 Statistics and Probability Probability and statistics are two related but separate academic disciplines. Statistical analysis often uses probability distributions and the two topics are often studied together. However, probability theory contains much that is of mostly of mathematical interest and not directly relevant to statistics. Moreover, many topics in statistics are independent of probability theory. Probability (or likelihood) is a measure or estimation of how likely it is that something will happen or that a statement is true. Probabilities are given a value between 0 (0% chance or will not happen) and 1 (100% chance or will happen). The higher the degree of probability, the more likely the event is to happen, or, in a longer series of samples, the greater the number of times such event is expected to happen. Statistics is the study of the collection, organization, analysis, interpretation, and presentation of data. It deals with all aspects of data, including the planning of data collection in terms of the design of surveys and experiments. Topics discussed in this chapter are as follows: ● Mean ● Median ● Mode ● Standard deviation ● Variance ● Coefficient of variation ● zScore ● Range ● Central limit theorem ● Counting rule for combinations ● Counting rule for permutations ● Binomial probability ● Poisson probability ● Confidence intervals ● Sample size ● Regression and correlation ● Pearson productmoment correlation coefficient ● Test statistic for hypothesis tests about a population proportion ● Chisquare goodnessoffit test statistic ● Standard normal distribution table ● Student’s tdistribution table ● Chisquare table ● Table of Fstatistics, P 5 0.05 7 Financial Mathematics The world of finance is literally FULL of mathematical models, formulas, and systems. It is absolutely necessary to understand certain key concepts in order to be successful financially, whether that means saving money for the future or to avoid being a victim of a quicktalking salesman. Financial mathematics is a collection of mathematical techniques that find application in finance, e.g., asset pricing: derivative securities, hedging and risk management, portfolio optimization, structured products. This chapter has links to math lessons about financial topics, such as annuities, savings rates, compound interest, and present value. Topics discussed in this chapter are as follows: ● Percentage ● The number of payments ● Convert interest rate compounding bases ● Effective interest rate ● The future value of a single sum ● The future value with compounding ● The future value of a cash flow series ● The future value of an annuity ● The future value of an annuity due ● The future value of an annuity with compounding ● Monthly payment ● The present value of a single sum ● The present value with compounding ● The present value of a cash flow series ● The present value of an annuity with continuous compounding ● The present value of a growing annuity with continuous compounding ● The net present value of a cash flow series ● Expanded net present value formula ● The present worth cost of a cash flow series ● The present worth revenue of a cash flow series Symbols used in financial mathematics are as follows: P: amount borrowed N: number of periods B: balance g: rate of growth m: compounding frequency r: interest rate rE: effective interest rate rN: nominal interest rate PMT: periodic payment FV: future value PV: present value CF: cash flow J: the jth period T: terminal or last period
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