A few words from the author . . .
Over the past three years I have tried to offer mathematical support to many hundreds of
students in the early stages of their degree programmes in engineering.
On many, many occasions I have found that gaps in mathematical knowledge impede progress
both in engineering mathematics and also in some of the engineering topics that the students are
studying. Sometimes these gaps arise because they have long-since forgotten basic techniques.
Sometimes, for a variety of reasons, they seem to have never met certain fundamentals in their
previous studies. Whatever the underlying reasons, the only practical remedy is to have available
resources which can quickly get to the heart of the problem, which can outline a technique or
formula or important results, and, importantly, which students can take away with them. This
Engineering Maths First-Aid Kit is my attempt at addressing this need.
I am well aware that an approach such as this is not ideal. What many students need is a
prolonged and structured course in basic mathematical techniques, when all the foundations
can be properly laid and there is time to practice and develop confidence. Piecemeal attempts
at helping a student do not really get to the root of the underlying problem. Nevertheless I
see this Kit as a realistic and practical damage-limitation exercise, which can provide sufficient
sticking plaster to enable the student to continue with the other aspects of their studies which
are more important to them.
I have used help leaflets similar to these in the Mathematics Learning Support Centre at Loughborough.
They are particularly useful at busy times when I may have just a few minutes to
try to help a student, and I would like to revise a topic briefly, and then provide a few simple
practice exercises. You should realise that these leaflets are not an attempt to put together a
coherent course in engineering mathematics, they are not an attempt to replace a textbook, nor
are they intended to be comprehensive in their treatment of individual topics. They are what I
say – elements of a First-Aid kit.
I hope that some of your students find that they ease the pain!
Tony Croft
December 1999
Author(s): Tony Croft
Publisher: Prentice Hall / Pearson
Year: 2000
Language: English
Pages: 168
1. Arithmetic 1.1.1
1.1 Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1
1.2 Powers and roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1
1.3 Scientific notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1
1.4 Factorials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1
1.5 The modulus of a number . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1
2. Algebra 2.1.1
2.1 The laws of indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1
2.2 Negative and fractional powers . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1
2.3 Removing brackets 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1
2.4 Removing brackets 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1
2.5 Factorising simple expressions . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1
2.6 Factorising quadratics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1
2.7 Simplifying fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1
2.8 Addition and subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.1
2.9 Multiplication and division . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.1
2.10 Rearranging formulas 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.1
2.11 Rearranging formulas 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11.1
2.12 Solving linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.12.1
2.13 Simultaneous equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13.1
2.14 Quadratic equations 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.14.1
2.15 Quadratic equations 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.15.1
2.16 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.16.1
2.17 The modulus symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.17.1
2.18 Graphical solution of inequalities . . . . . . . . . . . . . . . . . . . . . . . 2.18.1
2.19 What is a logarithm? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.19.1
2.20 The laws of logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.20.1
2.21 Bases other than 10 and e . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.21.1
2.22 Sigma notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.22.1
2.23 Partial fractions 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.23.1
2.24 Partial fractions 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.24.1
2.25 Partial fractions 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.25.1
2.26 Completing the square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.26.1
2.27 What is a surd? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.27.1
3. Functions, coordinate systems and graphs 3.1.1
3.1 What is a function? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1
3.2 The graph of a function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1
3.3 The straight line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1
3.4 The exponential constant e . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1
3.5 The hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1
3.6 The hyperbolic identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1
3.7 The logarithm function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1
3.8 Solving equations involving logarithms and exponentials . . . . . . . . . . . 3.8.1
3.9 Polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9.1
4. Trigonometry 4.1.1
4.1 Degrees and radians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1
4.2 The trigonometrical ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1
4.3 Graphs of the trigonometric functions . . . . . . . . . . . . . . . . . . . . . . 4.3.1
4.4 Trigonometrical identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1
4.5 Pythagoras’ theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1
4.6 The sine rule and cosine rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1
5. Matrices and determinants 5.1.1
5.1 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1
5.2 Cramer’s rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1
5.3 Multiplying matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1
5.4 The inverse of a 2 × 2 matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1
5.5 The inverse of a matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1
5.6 Using the inverse matrix to solve equations . . . . . . . . . . . . . . . . . . . 5.6.1
6. Vectors 6.1.1
6.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1
6.2 The scalar product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1
6.3 The vector product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1
7. Complex numbers 7.1.1
7.1 What is a complex number? . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1
7.2 Complex arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1
7.3 The Argand diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1
7.4 The polar form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1
7.5 The form r(cos θ + j sin θ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1
7.6 Multiplication and division in polar form . . . . . . . . . . . . . . . . . . . . 7.6.1
7.7 The exponential form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7.1
8. Calculus 8.1.1
8.1 Introduction to differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1
8.2 Table of derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1
8.3 Linearity rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1
8.4 Product and quotient rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1
8.5 The chain rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1
8.6 Integration as the reverse of differentiation . . . . . . . . . . . . . . . . . . . 8.6.1
8.7 Table of integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.1
8.8 Linearity rules of integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8.1
8.9 Evaluating definite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9.1
8.10 Integration by parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.10.1
8.11 Integration by substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.11.1
8.12 Integration as summation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.12.1
