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About this book :-
On the Shoulders of Giants A Course in Single Variable Calculus written by
G. Smith, G. Mclelland
In the history of the calculus of variations the existence of a minimizer was questioned only in the second half of the 19th century by Weierstraß. We present his famous counterexample to Dirichlet’s principle, which awakens the requirement for an existence theory. This leads to the “direct methods in the calculus of variations.” Here one independent variable has the advantage that the Sobolev spaces and the functional analytic tools can be given without great difficulties in the text or in the Appendix. Some emphasis is put on quadratic functionals, since their EulerLagrange equations are linear. The above-mentioned Dirichlet’s principle offers an elegant way to prove the existence of solutions of (linear) boundary value problems: simply obtain minimizers.
This clear and concise textbook provides a rigorous introduction to the calculus of variations, depending on functions of one variable and their first derivatives. It is based on a translation of a German edition of the book Variationsrechnung (Vieweg+Teubner Verlag, 2010), translated and updated by the author himself. Topics include: the Euler-Lagrange equation for one-dimensional variational problems, with and without constraints, as well as an introduction to the direct methods. The book targets students who have a solid background in calculus and linear algebra, not necessarily in functional analysis. Some advanced mathematical tools, possibly not familiar to the reader, are given along with proofs in the appendix. Numerous figures, advanced problems and proofs, examples, and exercises with solutions accompany the book, making it suitable for self-study.
The book Shoulders of Giants will be particularly useful for beginning graduate students from the physical, engineering, and mathematical sciences with a rigorous theoretical background. This book is an innovative introductory mathematics text for science & engineering students. The book presents the mathematical content of a traditional first year calculus course ; however, the subject matter is developed by modelling real life problems to demonstrate the way that mathematics pervades many aspects of our lives. The approach taken: 1/ Understands that mathematics should be introduced in a context tailored to its audience. 2/ Begins with a series of real life problems - an amusement park ride, an air disaster & problems caused by vibrating cables on a bridge. 3/ Uses the solutions to these problems as a means of providing a way of introducing mathematical concepts as they are needed & in a context suitable to the needs of that audience." Presents a treatment of single variable Calculus designed as an introductory tertiary level mathematics textbook for engineering and science students. The subject matter is developed by modeling physical problems, some of which would normally be encountered by students as experiments in a first year physics course.
Book Detail :-
Title: On the Shoulders of Giants A Course in Single Variable Calculus
Edition:
Author(s): G. Smith, G. Mclelland
Publisher: UNSW Press
Series:
Year: 2002
Pages: 291
Type: PDF
Language: English
ISBN: 0868407178,9780868407173
Country: Astralia
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About Author :-
Author G. J. McLelland is an Astralian Mathematician. He is professor from University of Technology, Sydney.
Author G. Smith is also Math expert.
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Book Contents :-
On the Shoulders of Giants A Course in Single Variable Calculus written by
G. Smith, G. Mclelland
cover the following topics.
1. Terror, tragedy and bad vibrations
1.1 Introduction
1.2 The Tower of Terror
1.3 Intothinair
1.4 Music andthebridge
1.5 Discussion
1.6 Rulesofcalculation
2. Functions
2.1 Rulesofcalculation
2.2 Intervalsonthe real line
2.3 Graphsof functions
2.4 Examplesof functions
3. Continuity and smoothness
3.1 Smoothfunctions
3.2 Continuity
4. Differentiation
4.1 Thederivative
4.2 Rules fordifferentiation
4.3 Velocity, acceleration and rates of change
5. Falling bodies
5.1 The Tower of Terror
5.2 Solving differential equations
5.3 General remarks
5.4 Increasinganddecreasingfunctions
5.5 Extremevalues
6. Series and the exponential function
6.1 The airpressureproblem
6.2 Infiniteseries
6.3 Convergenceof series
6.4 Radiusof convergence
6.5 Differentiationofpower series
6.6 The chainrule
6.7 Properties of the exponential function
6.8 Solution of the air pressure problem
7. Trigonometric functions
7.1 Vibratingstrings andcables
7.2 Trigonometric functions
7.3 More on the sine and cosine functions
7.4 Triangles, circles and the number
7.5 Exact values of the sine and cosine functions
7.6 Other trigonometric functions
8. Oscillation problems
8.1 Second order linear differential equations
8.2 Complexnumbers
8.3 Complexseries
8.4 Complex roots of the auxiliary equation
8.5 Simple harmonic motion and damping
8.6 Forced oscillations
9. Integration
9.1 Another problem on the Tower of Terror
9.2 Moreonairpressure
9.3 Integralsandprimitivefunctions
9.4 Areas under curves
9.5 Area functions
9.6 Integration
9.7 Evaluationof integrals
9.8 The fundamental theorem of the calculus
9.9 The logarithmfunction
10. Inverse functions
10.1 The existenceof inverses
10.2 Calculatingfunctionvalues for inverses
10.3 The oscillation problem again
10.4 Inverse trigonometric functions
10.5 Other inverse trigonometric functions
11. Hyperbolic functions
11.1 Hyperbolic functions
11.2 Properties of the hyperbolic functions
11.3 Inverse hyperbolic functions
12. Methods of integration
12.1 Introduction
12.2 Calculationofdefinite integrals
12.3 Integration by substitution
12.4 Integrationbyparts
12.5 The method of partial fractions
12.6 Integrals with a quadratic denominator
12.7 Concludingremarks
13. A nonlinear differential equation
13.1 The energyequation
13.2 Conclusion
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