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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化學數學（一）

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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 ChemistryNational Sun Yat-sen University

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

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.

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

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

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

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

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.

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.

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

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)

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)