Introductory Mathematics: Applications and Methods: (Springer Undergraduate Mathematics Series)

This text for first-year students of applied mathematics may also be used as a bridge between school and university study. The theoretical content is firmly based but the focus is on techniques and applications. Throughout, there are many exercises of varying difficulty, with worked solutions provided. Basic differentiation and integration techniques taught at school level are presented as a refresher. Matrix algebra, vector algebra and complex numbers are then described at sufficient depth for the student to feel comfortable with them when they reappear later in the book. Subsequent chapters move onto real variable calculus, including partial differentiation and multiple integrals. The concluding chapter in differential equations considers applications taken from both physical and economic contexts.

Table of Contents:
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.