化學數學(一)
Sponsored Links
This presentation is the property of its rightful owner.
1 / 96

化學數學(一) PowerPoint PPT Presentation


  • 94 Views
  • Uploaded on
  • Presentation posted in: General

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

Download Presentation

化學數學(一)

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


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.


Units (base)


Units (derived)


Metric Prefixes


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:


Limit as the Core of Modern Mathematics


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.


Differentiation of Elementary Functions


Common Rules for Differentiation


Frequently Used Derivatives


Implicit Function


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.


MacLaurin Series


Taylor Series


Approximation of Series

Taylor’s theorem:


l’Hôpital’s Rule


Approximation of Series


Integration as Limit of Sum


Common Rules for Integration

The fundamental theorem of the calculus:

The definite and indefinite integrals.


Elementary Integrals


y=f

x

b

a

y

x

Average of a Function


Integration of Odd/Even Functions


2

Special Case: Discontinuous Functions


c-ε

c+ε

a

b

c

Special Case: Improper Integrals


Special Case: Infinite 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


Constant volume


The Relation Between Cp and Cv

(perfect gas)


Common Rules for Partial Differentiation


s

y

θ

x

Change of Variables(Coordinate Transform)


Higher Derivatives


Higher Derivatives


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:


The -1 Rule (Chain Rule)


Using The -1 Rule


Differential with Constraint

Example:


Example


Change of Independent Variables(Coordinate Transform)


Classroom Exercise


Exact Differentials


z

M1

M2

m2

s

M4

M3

m1

y

x

Stationary Points


Stationary Points


Optimization with Constraints(Method of Lagrange Multipliers)


Optimization with Constraints(Method of Lagrange Multipliers)


Secular Equation


y

B

C

ds

dy

dx

A

x

a

x

x+dx

b

Curvilinear Integrals


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?


Multiple Integrals


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


Change of Variables: General Cases


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


Separation of Variables


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


  • Login