Introductory Mathematics: Applications and Methods

This book is aimed at undergraduate students embarking on the first year of a modular mathematics degree course. It is a self-contained textbook making it ideally suited to distance learning and a useful reference source for courses with the traditional lecture/tutorial structure. The theoretical content is firmly based but the principal focus is on techniques and applications. The important aims and objectives are presented clearly and then reinforced using complete worked solutions within the text. There is a natural increase in difficulty and understanding as each chapter progresses, always building upon the basic elements. It is assumed that the reader has studied elementary calculus at Advanced level and is at least familiar with the concept of function and has been exposed to basic differentiation and integration techniques. Although these are covered in the book they are presented as a refresher course to jog the student's memory rather than to introduce the topic for the first time. The early chapters cover the topics of matrix algebra, vector algebra and com plex numbers in sufficient depth for the student to feel comfortable -when they reappear later in the book. Subsequent chapters then build upon the student's 'A' level knowledge in the area of real variable calculus, including partial differentiation and mUltiple inte grals. The concluding chapter on differential equations motivates the student's learning by consideration of applications taken from both physical and eco nomic contexts.

A book for students to read before university to give themselves a head-start - Completely self-contained, with solutions provided to exercises of varying difficulty - Alternatively, universities may choose to base their first- semester course on this book - Applications and examples given from both physics and economics

Contenu

1. Simultaneous Linear Equations.- 1.1 Introduction.- 1.2 The method of determinants.- 1.3 Gaussian elimination.- 1.4 Ill-conditioning.- 1.5 Matrices.- 2. Vector Algebra.- 2.1 Introduction.- 2.2 Algebraic representation.- 2.3 Linear independence.- 2.4 The scalar product.- 2.5 The vector product.- 2.6 Triple products.- 2.7 Differentiation of vector functions.- 3. Complex Numbers.- 3.1 Introduction.- 3.2 Algebra of complex numbers.- 3.3 Graphical representation.- 3.4 Polar form.- 3.5 Exponential form.- 3.6 De Moivre's theorem.- 4. Review of Differentiation Techniques.- 4.1 Introduction.- 4.2 Differentiation of standard functions,.- 4.3 Function of a function.- 4.4 The product rule.- 4.5 The quotient rule.- 4.6 Higher derivatives.- 4.7 Implicit differentiation.- 4.8 Logarithmic differentiation.- 5. Review of Integration Techniques.- 5.1 Introduction.- 5.2 Integration of standard functions.- 5.3 Integration by substitution.- 5.4 Integration using partial fractions.- 5.5 Integration by parts.- 6. Applications of Differentiation.- 6.1 Introduction.- 6.2 Functions:.- 6.3 Curve sketching.- 6.4 Optimisation.- 6.5 Taylor series.- 7. Partial Differentiation.- 7.1 Introduction.- 7.2 Notation.- 7.3 Total differentials.- 7.4 The total derivative.- 7.5 Taylor series for functions of two variables.- 7.6 Maxima, minima and saddlepoints.- 7.7 Problems with constraints:.- 7.8 Lagrange multipliers.- 8. Multiple Integrals.- 8.1 Double integrais.- 8.2 Triple integrais.- 9. Differential Equations.- 9.1 Introduction.- 9.2 First order differential equations.- 9.3 Second order equations.- Solutions to Exercises.