Introductory Mathematics: Applications and Methods

ISBN-13: 9783540761792 / Angielski / Miękka / 1998 / 226 str.

Gordon Marshall; G. S. Marshall;Marshall

Opis

This book is aimed at undergraduate students on their first year of a modular mathematics degree course. The important aims and objectives are presented clearly and reinforced using complete worked solution within the text. There is a natural increase in difficulty and understanding as each chapter progresses, always building upon the basic elements. The early chapters cover matrix algebra, vector algebra and complex numbers. Subsequent chapters cover such topics as real variable calculus, partial differentiation and multiple integrals. The concluding chapter motivates the students learning by consideration of applications taken from both physical and economic contexts.

Spis treści

1. Simultaneous Linear Equations.- 1.1 Introduction.- 1.2 The method of determinants.- 1.3 Gaussian elimination.- 1.4 Ill-conditioning.- 1.5 Matrices.- 1.5.1 Matrix notation.- 1.5.2 Matrix algebra.- 1.5.3 Inverse matrices.- 1.5.4 Matrix solution of linear equations.- 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.- 3.6.1 Roots of unity.- 3.6.2 Roots of complex numbers.- 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.- 5.4.1 Introduction.- 5.4.2 Resolving into partial fractions.- 5.4.3 Repeated factors.- 5.5 Integration by parts.- 6. Applications of Differentiation.- 6.1 Introduction.- 6.2 Functions:.- 6.2.1 Increasing and decreasing functions.- 6.2.2 Relative maxima and minima.- 6.2.3 Concavity:.- 6.2.4 Points of inflection.- 6.2.5 Absolute maxima and minima.- 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.- 7.6.1 Classification of critical points.- 7.7 Problems with constraints:.- 7.8 Lagrange multipliers.- 8. Multiple Integrals.- 8.1 Double integrais.- 8.1.1 Change of variable in double integrals.- 8.2 Triple integrais.- 9. Differential Equations.- 9.1 Introduction.- 9.2 First order differential equations.- 9.2.1 Equations of the form $$\frac{}{} = f\left( x \right)$$.- 9.2.2 Equations of the form mr$$\frac{}{} = ky$$.- 9.2.3 Variables separable.- 9.2.4 Integrating factors.- 9.3 Second order equations.- 9.3.1 Linear equations with constant coefficients.- 9.3.2 Solution of the homogenous equation.- 9.3.3 Simple harmonic motion.- 9.3.4 Damped SHM.- 9.3.5 Particular integrals involving elementary functions.- Solutions to Exercises.