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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Mathematics for Engineers serves as a comprehensive guide on the basics of mathematics and their applications in engineering for students and seasoned professionals alike.
Mathematics for Engineers is designed to help students develop mathematical proficiencies, which are required in technical courses and careers involving strategic mathematical competence and adaptive reasoning. This volume also acts as a reference for professionals in engineering who need a refresher for their technical math skills. Through this book, students and professionals in the engineering discipline will build a capacity and expand their fundamental mathematical skills for logical thought, reflection, explanation, and justification in the field of applied science. This book is designed for general use for science and engineering students across the globe.
The book effectively compiles important information in one place alongside examples and practice problems with application and practice based questions. Emphasis is placed on the application of mathematics in appropriate context and modeling of real-world situations. By the end of this book, students and professionals in the engineering discipline will be able to present and process their mathematical reasoning and conclusions numerically, graphically, symbolically, and verbally.
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Veröffentlichungsjahr: 2025
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Table of Contents
Cover
Series Page
Title Page
Copyright Page
Preface
1 Fractions, Decimals, and Percentages
1.1 Introduction
1.2 Fractions
1.3 Decimals and Percentages
1.4 Check Your Knowledge (MCQ)
1.5 Activity/Experiment
2 Ratio and Proportion
2.1 Introduction
2.2 Ratios
2.3 Rate
2.4 Proportions
2.5 Activity/Experiment—Ratio
2.6 Activity/Experiment—Proportions
3 Two-Dimensional Geometry
3.1 Introduction
3.2 Geometry
3.3 Area and Perimeter of 2D Shapes
3.4 Curved Plane Figures
3.5 Perimeter and Area of Irregular Polygons and Composite Shapes
3.6 Check Your Knowledge (MCQ)
3.7 Activity/Experiment
4 Three-Dimensional Geometry
4.1 Introduction
4.2 Nets of Three-Dimensional Shapes
4.3 Surface Area and Volume of Three-Dimensional Shapes
4.4 Compounded Shapes
4.5 Check Your Knowledge (MCQ)
4.6 PBL/Experiment
5 Algebra and Graphs
5.1 Introduction
5.2 Algebraic Expressions
5.3 Transposition of Formulae
5.4 Algebraic Fractions
5.5 Simple Linear Equation
5.6 Quadratic Equations
5.7 Simultaneous Equations
5.8 Graphs of Algebraic Functions
5.9 Exponential Functions
5.10 Check Your Knowledge (MCQ)
5.11 Activity/Experiment
6 Exponent and Logarithm
6.1 Introduction
6.2 Indices
6.3 Logarithms
6.4 Check Your Knowledge (MCQ)
7 Trigonometry
7.1 Introduction
7.2 Pythagoras Theorem
7.3 Trigonometric Ratio
7.4 Trigonometry Table
7.5 The Angle of Elevation and Angle of Depression
7.6 Trigonometric Equation
7.7 Radian Measure
7.8 The Sine and Cosine Rules
7.9 Steps to Use Calculator for Trigonometry
7.10 Check Your Knowledge (MCQ)
8 Differential Calculus
8.1 Introduction
8.2 Functions
8.3 Limit and Continuity
8.4 Derivatives
8.5 Differentiation Rules
8.6 Derivatives of Logarithmic and Exponential Function
8.7 Derivatives of Trigonometric Function
8.8 Application of Derivatives
8.9 Maxima and Minima
8.10 Check Your Knowledge (MCQ)
8.11 Activity/Experiment
9 Integral Calculus
9.1 Introduction
9.2 Integration Rules
9.3 Integration by Parts
9.4 Integration by Partial Fraction
9.5 Definite Integral
9.6 Check Your Knowledge (MCQ)
10 Probability
10.1 Introduction
10.2 Fundamental Principle of Counting
10.3 Permutations and Combinations
10.4 Simple Probability
10.5 Conditional Probability
10.6 Probability Sample Space Diagram
10.7 Probability Tree Diagram
10.8 Probability Using Venn Diagram
10.8 Check Your Knowledge (MCQ)
10.9 PBL/Experiment (Experimental Probability Vs. Theoretical Probability)
Answers
About the Authors
Index
Also of Interest
End User License Agreement
Guide
Cover Page
Series Page
Title Page
Copyright Page
Preface
Table of Contents
Begin Reading
Answers
About the Authors
Index
Also of Interest
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Scrivener Publishing100 Cummings Center, Suite 541JBeverly, MA 01915-6106
Publishers at ScrivenerMartin Scrivener ([email protected])Phillip Carmical ([email protected])
Mathematics for Engineers
Ritu Shrivastava
Ramakant Bhardwaj
and
Satyendra Narayan
This edition first published 2025 by John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA and Scrivener Publishing LLC, 100 Cummings Center, Suite 541J, Beverly, MA 01915, USA© 2025 Scrivener Publishing LLCFor more information about Scrivener publications please visit www.scrivenerpublishing.com.
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Library of Congress Cataloging-in-Publication Data
ISBN 9781394214587
Front cover images supplied by Adobe FireflyCover design by Russell Richardson
Preface
Mathematics without practice is incomplete. Practice of mathematics without understanding theory is meaningless. Theory and practice are two sides of the same coin. Like the two sides of a coin, theory and practice have their differences, and they are inseparable. Mathematics for Engineers is designed to explain theory and practice together in a meaningful way, providing engineers with a comprehensive understanding of fundamental mathematical concepts crucial for their academic and professional endeavors. The depth and breadth of this textbook offers comprehensive coverage of all the basic mathematical tools needed for first-year engineering students, as well as a reference for veteran engineers and industry professionals.
This textbook is originated from a one-semester course dedicated to introductory engineering mathematics taught in many universities and engineering colleges over the past several years primarily to first-year engineering students. This covers a wide range of syllabus requirements globally. In particular, the textbook is highly suitable for the latest National Certificate and Diploma courses and Vocational Certificate of Education in science & engineering programs. First year undergraduates who need some remedial mathematics may also find it helpful in meeting their needs. Overall, the textbook provides a solid foundation in mathematical principles, which may enable students to solve mathematical, scientific, and associated engineering principles. Key Features of this book are highlighted below.
Chapter 1: Fractions, Decimals, and Percentages lays the groundwork by revisiting fundamental arithmetic concepts, such as fractions, decimals, and percentages. From understanding the basics of fractions to performing algebraic operations and mental math tricks, it equips leaners with essential skills for more complex mathematical endeavors.
Chapter 2: Ratios and Proportions exploring ratios, rates, and proportions delves into the principles governing the relative magnitudes of quantities. They play pivotal roles in engineering, influencing everything from structural design to financial analysis. Learners will learn techniques for simplifying ratios, finding equivalent ratios, and solving problems involving direct and inverse proportions.
Chapter 3 and 4: 2D & 3D Geometry are fundamental to engineering, and it covers essential concepts related to 2D and 3D shapes, perimeter, area, surface area, and volume calculations. Understanding geometric principles is important, crucial, and required for designing structures and analyzing spatial relationships in engineering applications.
Chapter 5: Algebra and Graphs provides algebraic skills that are indispensable for engineers. It covers equations, expressions, and graphical representations of mathematical relationships. From linear to quadratic equations and beyond, learners will explore various algebraic techniques required for problem-solving in engineering contexts.
Chapter 6: Exponentials & Logarithms are the powerful mathematical tools. They are used in a wide range of applications in engineering problems. It elucidates the rules governing these concepts, their graphical representations, and their role in modeling growth and decay phenomena.
Chapter 7: Trigonometry is indispensable for analyzing angles, distances, and relationships within geometric structures. It covers trigonometric ratios, Pythagoras’ theorem, trigonometric equations, and applications such as finding angles of elevation and depression.
Chapter 8 and 9: Differential Calculus & Integration provide engineers with powerful tools for analyzing rates of change, optimization problems, and modeling dynamic systems. Differentiation, integration, and their applications in engineering contexts are covered completely.
Chapter 10: Probability plays an important and crucial role in engineering decision-making problems and risk assessment. This chapter introduces principles of counting, permutations, combinations, and various techniques for calculating probabilities of single and compound events.
Each chapter is dedicated to building a strong foundation, engineering applications, problem-solving skills, culturally relevant context, and practical problem-based learning (PBL) exercises and experiments to reinforce learner's understanding and application of mathematical concepts in real-world engineering scenarios.
The authors hope this textbook serves as a valuable resource in the student's journey to mastering mathematics for engineering and as a reference for the veteran engineer or industry professional. It does not matter whether you are just embarking on your academic studies or a practicing engineer seeking to enhance your mathematical skills and proficiency. This book is designed to support each individual's educational and professional development.
The authors of this book would like to acknowledge the use of freely available mathematical figures and images sourced from various online repositories. These resources have been invaluable in illustrating key concepts and enhancing the visual appeal of this book.
This book builds upon the foundational work of many brilliant mathematicians. Their groundbreaking discoveries have shaped our understanding of the world and continue to inspire generations of researchers.
2Ratio and Proportion
Objectives: This chapter
Demonstrate how to express ratios as fractions and rates as fractions.
Determine proportions.
Solve problems by writing proportions.
Apply ratios and proportions to solve real-life problems.
2.1 Introduction
Every day in our lives, we need to compare quantities. We use the ratio to compare every field, such as homes, businesses, sports, science, engineering, etc.
There are a few examples of the ratio.
The football team won 5 games for every 3 games lost.
The employee gets a three-monthly bonus every two years.
In simple words, we can say that ratios are another way of demonstrating fractions.
2.2 Ratios
2.2.1 Definition
The relationship between two quantities is called a ratio and is generally expressed as the quotient of one divided by the other.
Note 1: To find any ratio, do we need to ensure that the quantities are in the same unit? If not, we need to convert it by using an appropriate rule.
Note 2:We can write ratio in three different ways. like the ratio of 1 to 2 can be written as
1 is to 2 or 1: 2 or
Remember, no matter which notation we use for this ratio; it is read as “1 is to 2.”
2.2.2 Simplifying Ratio
We already learned about the simplest form of any fraction in the previous chapter.
If a fraction’s numerator and denominator share only one common factor, it is said to be in its simplest form.
The same will apply to the ratio.
For example: Write the ratio 50 to 100 in the simplest form.
SOLUTION: The ratio 50 to 100 can be expressed as a fraction
So, the ratio of 50 to 100 is the same as that of 1 to 2.
2.2.3 Equivalent Ratio
The ratio of 3:2 can also be written as 6:4.
Two ratios are equivalent if the fraction corresponding to them is equivalent.
For example: to are equivalent ratios since they represent the same fraction.
2.2.4 Methods for Finding a Ratio
Unitary method: In this method, first, we find the value of one unit or the value of one part and then calculate the value of the required number of units or parts.
Problem type 1: Where ratio and total amount are given.
EXAMPLE:
Divide BHD 16 into 3: 5.
Step 1:
Let the two ratios be 3
x
and 5
x
.
Step 2:
Given that the total ratio is equal to BHD 16.
Step 3:
Therefore, 2
x
+ 7
x
= 18 = 9
x
= 18
Step 4:
So that the value of one-part equals 2 BHD. Multiply by the ratio numbers
Step 5:
Check your answers. Values from Step 4 should be added to the correct amount. 6+10 = 16
Scholar’s style:Total given
Total ratio = 3+5 = 8
Value of one part = = 2 BD
Value of 3 parts = 3 × 2 = 6BD
Value of 5 parts = 5 × 2 = 10BD
Total 6BD + 10 BD = 16BD
Problem type 2: Where total is not given, but the ratio and the amount of any one ratio are provided.
EXAMPLE:
Divide the money into two parts in the ratio of 3:7. If the large part is BHD 735, find the total amount.
Step 1:
Let the two ratios be 3
x
and 7
x
.
Step 2:
Given that 7
x
(larger part) = 735
Step 3: T
herefore,
Step 4:
So that the value of one-part equals 105.
Multiply this value with the remaining ratio 3 × 105 = 315
Step 5:
Add up each amount to get Total = 735 + 315 = 1050 BHD
Therefore, the total amount is =1050 BHD
Scholar’s style:Total = unknown
Given ratio 3: 7
Given the value of 7 parts 7x= 735
So the value of one part x = 105
Multiply this value by the remaining ratio
3 × 105 = 315 (value of 3 parts)
Add up both amount = 315 + 735 = 1050 BD
2.2.5 Methods to Calculate Increasing and Decreasing in Each Ratio
For any given ratio, it is easy to calculate the increase and the decrease by just multiplying or dividing the same number of both ratios, have a look at the given example.
In a recipe for tea for eight people, we need 2 cups of milk, 4 spoons of sugar, 4 spoons of tea powder, ½ spoon of cardamom powder and ½ spoon of ginger powder. How much of each component is required to make tea for 24 people using the ratio of ingredients provided?
SOLUTION: Find what each amount needs to be multiplied (or divided) by. In this case, tea for 8 people must be made for 24 people. Since 8×3 = 24
So, everything needs to be multiplied by 3.
Step 2:
Ingredients
8 people
24 people
Milk
2
6
Sugar
4
12
Tea powder
4
12
Cardamom
½
1 ½
Ginger powder
½
1 ½
EXAMPLE 1:
Express the ratio 90: 180 in the simplest form.
SOLUTION: To express any ratio in its simplest form, we need to divide both ratios by their HCF (highest common factor)
Let’s find the HCF of both ratios.
Factors of 90 are 1,2,3,5,6,9,10,15,18,30,45,90.
Factors of 180 are 1,2,3,4,5,6,9,10,12,15,18,30,36,45,60,90, 180.
We can observe that 90 and 180 have a 90 HCF (Highest Common Factor).
Therefore,
Hence the simplest form of 90 and 180 is 1:2.
EXAMPLE 2:
Sameer and Sarah got 6 and 8 marks on a math test. What is the ratio of their marks in the simplest form?
SOLUTION: Given that, the ratio of marks = 6:8
Let’s find the HCF of both ratios.
Factors of 6 are 1,2,3,6
Factors of 8 are 1,2,4,8
The Highest Common Factor (HCF) of 6 and 8 is 2.
Therefore,
Hence the simplest form of Sameer and Sarah’s marks in the math’s test is 3:4.
EXAMPLE 3:
Divide 660 cm into 1: 2: 3.
SOLUTION: Add up the ratio number → 1+2+3 = 6
Multiply by the ratio number.
Check that your answer adds to the correct number.
EXAMPLE 4:
In a triangle, all three angles are in a ratio of 1:2:3. Suppose the largest angle is 90°. Find the other two angles.
SOLUTION: Divide the given angle by the corresponding ratio
Add up all angles to check your answer 30° + 60° + 90° = 180°
EXAMPLE 5:
Three numbers are in a ratio of 2:3:5. If the total of these three numbers is 200, find the numbers.
SOLUTION: Since the given ratio is 2:3:5. So the three numbers are 2x, 3x, and 5x,
Thus, we must divide 200 into three parts, 2x, 3x, and 5x.
Therefore, 2x +3x +5 x = 200
10x = 200
To check your answer, add all numbers 40 + 60 + 100 =200.
Level 1: Exercise 2.1 Practice Questions
In a mall, the ratio of female to male students is 15:25; express the ratio in its simplest form.
Aisha did 25 hours of work, and Sana did 75 hours of work. Find the ratio of Aisha’s work to Sana’s work.
Write the following ratios in their simplest form.
12: 48
9: 36
3: 6: 12
55: 77
500 L: 4000 L
70 kg: 350 kg
3: 21: 90
6 km: 1250 km
16 ml: 3 L
1600 km: 2 km
BHD 7: 7000 fills [ 1 BHD = 1000fills]
(l) 1000 g:
