Introduction to Mathematics for Life Scientists
1st Edition
Published on 6. December 2012
509 pages
978-3-642-96080-2 (ISBN)
Description
A few decades ago mathematics played a modest role in life sciences. Today, however, a great variety of mathematical methods is applied in biology and medicine. Practically every mathematical procedure that is useful in physics, chemistry, engineering, and economics has also found an important application in the life sciences. The past and present training of life scientists does by no means reflect this development. However, the impact of the fast growing number of applications of mathematical methods makes it indispensable that students in the life sciences are offered a basic training in mathematics, both on the undergraduate and the graduate level. This book is primarily designed as a textbook for an introductory course. Life scientists may also use it as a reference to find mathematical methods suitable to their research problems. Moreover, the book should be appropriate for self-teaching. It will also be a guide for teachers. Numerous references are included to assist the reader in his search for the pertinent literature.
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E. Batschelet
Introduction to Mathematics for Life Scientists
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Edward Batschelet
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Content
1. Real Numbers.- 1.1 Introduction.- 1.2 Classification and Measurement.- 1.3 A Problem with Percentages.- 1.4 Proper and Improper Use of Percentages.- 1.5 Algebraic Laws.- 1.6 Relative Numbers.- 1.7 InequaHties.- 1.8 Mean Values..- 1.9 Summation.- 1.10 Powers.- 1.11 Fractional Powers.- 1.12 Calculations with Approximate Numbers.- *1.13 An Application.- 1.14 Survey.- Problems for Solution.- 2. Sets and Symbolic Logic.- 2.1 "New Mathematics".- 2.2 Sets.- 2.3 Notations and Symbols.- 2.4 Variable Members.- 2.5 Complementary Set.- 2.6 The Union.- 2.7 The Intersection.- *2.8 Symbolic Logic.- *2.9 Negation and Implication.- *2.10 Boolean Algebra.- Problems for Solution.- 3. Relations and Functions.- 3.1 Introduction.- 3.2 Product Sets.- 3.3 Relations.- 3.4 Functions.- 3.5 A Special Linear Function.- 3.6 The General Linear Function.- *3.7 Linear Relations.- Problems for Solution.- 4. The Power Function and Related Functions.- 4.1 Definitions.- 4.2 Examples of Power Functions.- 4.3 Polynomials.- 4.4 Differences.- 4.5 An Application.- 4.6 Quadratic Equations.- Problems for Solution.- 5. Periodic Functions.- 5.1 Introduction and Definition.- 5.2 Angles.- 5.3 Polar Coordinates.- 5.4 Sine and Cosine.- 5.5 Conversion of Polar Coordinates.- 5.6 Right Triangles.- 5.7 Trigonometric Relations.- *5.8 Polar Graphs.- *5.9 Trigonometric Polynomials Ill.- Problems for Solution.- 6. Exponential and Logarithmic Functions I.- 6.1 Sequences.- 6.2 The Exponential Function.- 6.3 Inverse Functions.- 6.4 The Logarithmic Functions.- 6.5 Applications.- *6.6 Scaling.- *6.7 Spirals.- Problems for Solution.- 7. Graphical Methods.- 7.1 Nonlinear Scales.- 7.2 Semilogarithmic Plot.- 7.3 Double-Logarithmic Plot.- *7.4 Triangular Charts.- *7.5 Nomography.- *7.6 Pictorial Views.- Problems for Solution.-Chapters.8. Limits.- 8.1 Limits of Sequences.- 8.2 Some Special Limits.- 8.3 Series.- 8.4 Limits of Functions.- *8.5 The Fibonacci Sequence.- Problems for Solution.- 9. Differential and Integral Calculus.- 9.1 Growth Rates.- 9.2 Differentiation.- 9.3 The Antiderivative.- 9.4 Integrals.- 9.5 Integration.- 9.6 The Second Derivative.- 9.7 Extremes.- 9.8 Mean of a Continuous Function.- 9.9 Small Changes.- Problems for Solution.- 10. Exponential and Logarithmic Functions II.- 10.1 Introduction.- 10.2 Integral of 1/x.- 10.3 Properties of Injc.- 10.4 The Inverse Function of Inx.- 10.5 The General Definition of a Power.- 10.6 Relationship between Natural and Common Logarithms.- 10.7 Differentiation and Integration.- 10.8 Some Limits.- 10.9 Applications.- 10.10 Approximations and Series Expansions.- *10.11 Hyperbolic Functions.- Problems for Solution.- 11. Ordinary Differential Equations.- 11.1 Introduction.- 11.2 Geometric Interpretation.- 11.3 The Differential Equation y' = ay.- 11.4 The Differential Equation y'= ay+b.- 11.5 The Differential Equation y' = ay2+by+c.- 11.6 The Differential Equation dy/dx = k.y/x.- 11.7 A System of Linear Differential Equations.- 11.8 A System of Nonlinear Differential Equations.- * 11.9 Classification of Differential Equations.- Problems for Solution.- 12. Functions of Two or More Independent Variables.- 12.1 Introduction.- 12.2 Partial Derivatives.- 12.3 Maxima and Minima.- *12.4 Partial Differential Equations.- Problems for Solution.- 13. Probability.- 13.1 Introduction.- 13.2 Events.- 13.3 The Concept of Probability.- 13.4 The Axioms of Probability Theory.- 13.5 Conditional Probability.- 13.6 The Multiplication Rule.- 13.7 Counting.- 13.8 Binomial Distribution.- *13.9 Random Variables.- *13.10 The Poisson Distribution.- * 13.11Continuous Distributions.- Problems for Solution.- 14. Matrices and Vectors.- 14.1 Notations.- 14.2 Matrix Algebra.- 14.3 Applications.- 14.4 Vectors in Space.- 14.5 Applications.- Problems for Solution.- 15. Complex Numbers.- 15.1 Introduction.- 15.2 The Complex Plane.- 15.3 Algebraic Operations.- 15.4 Exponential and Logarithmic Functions of Complex Variables.- 15.5 Quadratic Equations.- 15.6 Oscillations.- Problems for Solution.- Answers to Problems.- References.- Author and Subject Index.
