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） 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.
Atomic Units Table: Basic quantities for the atomic unit system
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)
Polynomials Factorization: Roots (zeros of f(x))
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.
Common Finite Series Classroom exercise: Prove any of above sums
Important Infinite Series Classroom exercise: Prove
Convergence and Divergence (unbelievable billionaire!) Necessary for convergence: Further test of convergence: By comparison: d’Alembert’s ratio test:
Find the Limit of a Function (Classroom exercise)
y=f(x) x Differentiation as Limit of Division
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.
Successive Differentiation How about odd n? (Classroom exercise)
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)
Maxwell-Boltzmann Distribution of Speed (Classroom exercise: Verify the expression for the most probable speed.) v*
Consecutive elementary reactions Classroom exercise: Find the maximum of species I.
Approximation of Series Taylor’s theorem:
Common Rules for Integration The fundamental theorem of the calculus: The definite and indefinite integrals.
y=f x b a y x Average of a Function
2 Special Case: Discontinuous Functions
c-ε c+ε a b c Special Case: Improper Integrals
Example Calculate the mean speed of N2 at 25 oC
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)
z P z y O x y x Functions of Several Variables
z P z y O x y x Partial Differentiation