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化學數學(一). The Mathematics for Chemists (I) (Fall Term, 2004) (Fall Term, 2005) (Fall Term, 2006) Department of Chemistry National Sun Yat-sen University. Chapter 1 Review of Calculus. Numbers and variables Units Algebraic, transcendental, complex functions Coordinate systems Limit

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    1. 化學數學(一) The Mathematics for Chemists (I) (Fall Term, 2004)(Fall Term, 2005)(Fall Term, 2006)Department of ChemistryNational Sun Yat-sen University

    2. Chapter 1 Review of Calculus • Numbers and variables • Units • Algebraic, transcendental, complex functions • Coordinate systems • Limit • Differentiation and derivative • Integration • Series expansion Assignment for Chapter 1 : p.92:74 p.122: 32 pp. 142-144: 18,37,45,51,66,71 pp.168-170: 31,34,62,68,70 p.188: 34,49,55,56 pp.224-226: 18,25,33/36,50,53,55 p.241: 19, 28

    3. Numbers Integers (natural, whole, positive, negative, even, odd, composite, prime) Real numbers: rational irrational (surds, transcendental)) fixed point and floating point Complex numbers The discover (Hippasus) of first irrational number was thrown into sea.

    4. Units (base)

    5. Units (derived)

    6. Metric Prefixes

    7. Atomic Units Table: Basic quantities for the atomic unit system

    8. Table: Quantities for the atomic unit system

    9. Variables, Algebra and Functions • Continuous vs discrete • Domain (of definition, of value) a+b=b+a, ab=ba (commutative) a+(b+c)=(a+b)+c, (ab)c=a(bc), (associative) a(b+c)=ab+ac (distributive)

    10. Polynomials Factorization: Roots (zeros of f(x))

    11. f x x0 Rational Functions Singularity(奇點): (Here the roots of P(x) are the singularities of f(x))

    12. Transcendental Functions • Trigonometric functions • Inverse trigonometric functions • The exponential function • The logarithmic function • Hyperbolic functions

    13. Classroom Exercise • Write the singularities of the following functions (if they exist!):

    14. y r θ x Complex Functions (Proof) (classroom exercise)

    15. Proof (by Mathematical Induction) Recall the properties of trigonometric functions Why is mathematical induction valid and exact? An equation is worth infinite number of data; a proof infinite number of examples.

    16. Common Finite Series Classroom exercise: Prove any of above sums

    17. Important Infinite Series Classroom exercise: Prove

    18. Convergence and Divergence (unbelievable billionaire!) Necessary for convergence: Further test of convergence: By comparison: d’Alembert’s ratio test:

    19. Limit as the Core of Modern Mathematics

    20. Find the Limit of a Function (Classroom exercise)

    21. y=f(x) x Differentiation as Limit of Division

    22. Mysterious Infinitesimal What is dx? It is a variable. It can be as small as required. Its limit is zero, but it is absolutely not the same as zero. The existence of dx relies on a great property (continuity) of real numbers. The discovery of infinitesimal is one of the greatest discoveries in science.

    23. Differentiation of Elementary Functions

    24. Common Rules for Differentiation

    25. Frequently Used Derivatives

    26. Implicit Function

    27. Successive Differentiation How about odd n? (Classroom exercise)

    28. A B C F D E A B C Stationary Points A,B,C A Turning points C B Local minima: E,C Global minimum: C Local maxima: A, D, F Global maximum: D

    29. Q θ1 r1 y1 O Phase boundary x2 x1 y2 θ2 r2 P Snell’s Law of Refraction To find point O so that the time used for the light to travel from P to Q is minimized. (Principle of least time)

    30. Maxwell-Boltzmann Distribution of Speed (Classroom exercise: Verify the expression for the most probable speed.) v*

    31. Consecutive elementary reactions Classroom exercise: Find the maximum of species I.

    32. MacLaurin Series

    33. Taylor Series

    34. Approximation of Series Taylor’s theorem:

    35. l’Hôpital’s Rule

    36. Approximation of Series

    37. Integration as Limit of Sum

    38. Common Rules for Integration The fundamental theorem of the calculus: The definite and indefinite integrals.

    39. Elementary Integrals

    40. y=f x b a y x Average of a Function

    41. Integration of Odd/Even Functions

    42. 2 Special Case: Discontinuous Functions

    43. c-ε c+ε a b c Special Case: Improper Integrals

    44. Special Case: Infinite Integrals

    45. Example Calculate the mean speed of N2 at 25 oC

    46. Molecular Interpretation of Internal Energy • Equipartition theorem: at temperature T, the average of each quadratic contribution to the energy is the same and equal to kBT/2. (Monatomic gas)

    47. z P z y O x y x Functions of Several Variables

    48. z P z y O x y x Partial Differentiation

    49. Constant volume