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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

### 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

Table: 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

### 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

2

c-ε

c+ε

a

b

c

### 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

z

P

z

y

O

x

y

x

Constant volume

(perfect gas)

s

y

θ

x

z

R

Q

P

p

r

z

q

y

O

Δx

Δy

x

### Total Differential

Change of Internal Energy as a Total Differential

Volume as a Total Differential

### The Total Derivative

Classroom exercise:

Example:

Example

z

M1

M2

m2

s

M4

M3

m1

y

x

y

B

C

ds

dy

dx

A

x

a

x

x+dx

b

y

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.

2

(p2,T2)

p

1

1

(p1,T1)

0

T

### Entropy: A State Function

Which path is easier for us to calculate?

y

2

(1,1)

y=x

y=x2

R

x

(0,0)

### Change of Variables

2D proof of integration with changed variables

dB

dA

(x,y)

r

y

θ

x

Classroom exercise:

Finish the last step.

y

r

r+dr

x

dxdy = |J|rdrdθ

z

z

P(r,θ, )

P(x,y,z)

y

y

O

z

x

x

y

x

z

P(ρ, ,z)

y

x

### General Curvilinear Coordinates

Cylindrical polar coordinates:

Classroom exercise: write the volume element

In cylindrical polar coordinates.