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 |  موضوع: كتاب Mathematical Formulas for Industrial and Mechanical Engineering الثلاثاء 11 يناير 2022, 2:22 am | |
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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 (two-dimensional), 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 well-defined
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
● First-order differential equation
● Second-order 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
● z-Score
● 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
● Chi-square goodness-of-fit test statistic
● Standard normal distribution table
● Student’s t-distribution table
● Chi-square table
● Table of F-statistics, 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 quick-talking 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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