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 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

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.


Units base

Units (base)


Units derived

Units (derived)


Metric prefixes

Metric Prefixes


Atomic units

Atomic Units

Table: Basic quantities for the atomic unit system


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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:


Limit as the core of modern mathematics

Limit as the Core of Modern Mathematics


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.


Differentiation of elementary functions

Differentiation of Elementary Functions


Common rules for differentiation

Common Rules for Differentiation


Frequently used derivatives

Frequently Used Derivatives


Implicit function

Implicit Function


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.


Maclaurin series

MacLaurin Series


Taylor series

Taylor Series


Approximation of series

Approximation of Series

Taylor’s theorem:


L h pital s rule

l’Hôpital’s Rule


Approximation of series1

Approximation of Series


Integration as limit of sum

Integration as Limit of Sum


Common rules for integration

Common Rules for Integration

The fundamental theorem of the calculus:

The definite and indefinite integrals.


Elementary integrals

Elementary Integrals


Average of a function

y=f

x

b

a

y

x

Average of a Function


Integration of odd even functions

Integration of Odd/Even Functions


Special case discontinuous functions

2

Special Case: Discontinuous Functions


Special case improper integrals

c-ε

c+ε

a

b

c

Special Case: Improper Integrals


Special case infinite integrals

Special Case: Infinite 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


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Constant volume


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)


Higher derivatives

Higher Derivatives


Higher derivatives1

Higher Derivatives


Total differential

z

R

Q

P

p

r

z

q

y

O

Δx

Δy

x

Total Differential


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Change of Internal Energy as a Total Differential


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Volume as a Total Differential


The total derivative

The Total Derivative

Classroom exercise:


The 1 rule chain rule

The -1 Rule (Chain Rule)


Using the 1 rule

Using The -1 Rule


Differential with constraint

Differential with Constraint

Example:


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Example


Change of independent variables coordinate transform

Change of Independent Variables(Coordinate Transform)


Classroom exercise1

Classroom Exercise


Exact differentials

Exact Differentials


Stationary points1

z

M1

M2

m2

s

M4

M3

m1

y

x

Stationary Points


Stationary points2

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)


Secular equation

Secular Equation


Curvilinear integrals

y

B

C

ds

dy

dx

A

x

a

x

x+dx

b

Curvilinear Integrals


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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?


Multiple integrals

Multiple Integrals


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y

2

(1,1)

y=x

y=x2

R

x

(0,0)


Change of variables

Change of Variables


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2D proof of integration with changed variables

dB

dA

(x,y)


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r

y

θ

x

Classroom exercise:

Finish the last step.


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y

r

r+dr

x

dxdy = |J|rdrdθ


Change of variables general cases

Change of Variables: General Cases


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


Integrals in 3 dimensions

Integrals 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.


Fields scalar vector and tensor

Fields: Scalar, Vector and Tensor


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