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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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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
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
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
Atomic Units

Table: Basic quantities for the atomic unit system


Table: Quantities for the atomic unit system


Variables algebra and functions
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
Polynomials

Factorization:

Roots (zeros of f(x))


Rational functions

f

x

x0

Rational Functions

Singularity(奇點):

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


Transcendental functions
Transcendental Functions

  • Trigonometric functions

  • Inverse trigonometric functions

  • The exponential function

  • The logarithmic function

  • Hyperbolic functions


Classroom exercise
Classroom Exercise

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


Complex functions

y

r

θ

x

Complex Functions

(Proof) (classroom exercise)


Proof by mathematical induction
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
Common Finite Series

Classroom exercise:

Prove any of above sums


Important infinite series
Important Infinite Series

Classroom exercise:

Prove


Convergence and divergence
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
Find the Limit of a Function

(Classroom exercise)


Differentiation as limit of division

y=f(x)

x

Differentiation as Limit of Division


Mysterious infinitesimal
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
Successive Differentiation

How about odd n? (Classroom exercise)


Stationary points

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


Snell s law of refraction

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
Maxwell-Boltzmann Distribution of Speed

(Classroom exercise:

Verify the expression for the most probable speed.)

v*


Consecutive elementary reactions
Consecutive elementary reactions

Classroom exercise:

Find the maximum of species I.




Approximation of series
Approximation of Series

Taylor’s theorem:


L h pital s rule
l’Hôpital’s Rule




Common rules for integration
Common Rules for Integration

The fundamental theorem of the calculus:

The definite and indefinite integrals.



Average of a function

y=f

x

b

a

y

x

Average of a Function



Special case discontinuous functions

2

Special Case: Discontinuous Functions


Special case improper integrals

c-ε

c+ε

a

b

c

Special Case: Improper Integrals



Example
Example

Calculate the mean speed of N2 at 25 oC


Molecular interpretation of internal energy
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)


Functions of several variables

z

P

z

y

O

x

y

x

Functions of Several Variables


Partial differentiation

z

P

z

y

O

x

y

x

Partial Differentiation



The relation between c p and c v
The Relation Between Cp and Cv

(perfect gas)


Common rules for partial differentiation
Common Rules for Partial Differentiation


Change of variables coordinate transform

s

y

θ

x

Change of Variables(Coordinate Transform)




Total differential

z

R

Q

P

p

r

z

q

y

O

Δx

Δy

x

Total Differential




The total derivative
The Total Derivative

Classroom exercise:






Change of independent variables coordinate transform
Change of Independent Variables(Coordinate Transform)




Stationary points1

z

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)


Optimization with constraints method of lagrange multipliers1
Optimization with Constraints(Method of Lagrange Multipliers)



Curvilinear integrals

y

B

C

ds

dy

dx

A

x

a

x

x+dx

b

Curvilinear Integrals


y

B

1

x

A

0

1


Curvilinear integrals independent of path

(x2,y2)

B

y

1

A

(x1,y1)

0

x

Curvilinear Integrals Independent of Path

It depends on the initial and final coordinates only.


Entropy a state function

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)




r

y

θ

x

Classroom exercise:

Finish the last step.


y

r

r+dr

x

dxdy = |J|rdrdθ



Functions in 3 dimensions

z

z

P(r,θ, )

P(x,y,z)

y

y

O

z

x

x

y

x

Functions in 3 Dimensions



Separation of variables
Separation of Variables


General curvilinear coordinates

z

P(ρ, ,z)

y

x

General Curvilinear Coordinates

Cylindrical polar coordinates:

Classroom exercise: write the volume element

In cylindrical polar coordinates.



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