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The Mathematics for Chemists (I) (Fall Term, 2004)(Fall Term, 2005)(Fall Term, 2006)Department of ChemistryNational Sun Yat-sen University

• 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

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.

Table: Basic quantities for the atomic unit system

Table: Quantities for the atomic unit system

• 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)

Factorization:

Roots (zeros of f(x))

x

x0

Rational Functions

Singularity(奇點):

(Here the roots of P(x) are the singularities of f(x))

• Trigonometric functions

• Inverse trigonometric functions

• The exponential function

• The logarithmic function

• Hyperbolic functions

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

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.

Classroom exercise:

Prove any of above sums

Classroom exercise:

Prove

(unbelievable billionaire!)

Necessary for convergence:

Further test of convergence:

By comparison:

d’Alembert’s ratio test:

(Classroom exercise)

x

Differentiation as Limit of Division

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.

How about odd n? (Classroom exercise)

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

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)

(Classroom exercise:

Verify the expression for the most probable speed.)

v*

Classroom exercise:

Find the maximum of species I.

Taylor’s theorem:

l’Hôpital’s Rule

The fundamental theorem of the calculus:

The definite and indefinite integrals.

x

b

a

y

x

Average of a Function

Special Case: Discontinuous Functions

c-ε

c+ε

a

b

c

Special Case: Improper Integrals

Calculate the mean speed of N2 at 25 oC

• Equipartition theorem: at temperature T, the average of each quadratic contribution to the energy is the same and equal to kBT/2.

(Monatomic gas)

P

z

y

O

x

y

x

Functions of Several Variables

P

z

y

O

x

y

x

Partial Differentiation

The Relation Between Cp and Cv

(perfect gas)

Common Rules for Partial Differentiation

y

θ

x

Change of Variables(Coordinate Transform)

R

Q

P

p

r

z

q

y

O

Δx

Δy

x

Total Differential

Classroom exercise:

Example:

Change of Independent Variables(Coordinate Transform)

M1

M2

m2

s

M4

M3

m1

y

x

Stationary Points

Optimization with Constraints(Method of Lagrange Multipliers)

Optimization with Constraints(Method of Lagrange Multipliers)

B

C

ds

dy

dx

A

x

a

x

x+dx

b

Curvilinear Integrals

B

1

x

A

0

1

(x2,y2)

B

y

1

A

(x1,y1)

0

x

Curvilinear Integrals Independent of Path

It depends on the initial and final coordinates only.

(p2,T2)

p

1

1

(p1,T1)

0

T

Entropy: A State Function

Which path is easier for us to calculate?

2

(1,1)

y=x

y=x2

R

x

(0,0)

y

θ

x

Classroom exercise:

Finish the last step.

r

r+dr

x

dxdy = |J|rdrdθ

z

P(r,θ, )

P(x,y,z)

y

y

O

z

x

x

y

x

Functions in 3 Dimensions

Separation of Variables

P(ρ, ,z)

y

x

General Curvilinear Coordinates

Cylindrical polar coordinates:

Classroom exercise: write the volume element

In cylindrical polar coordinates.