1

課程名稱： 化 學 數 學 (一) Mathematics for Chemists本課程主要針對化學系同學在將來在化學相關課程中可能遇到的數學問題，舉凡微分方程、線性代數、向量分析、群論、統計學都用得上。本學期內容分兩大部分︰微分方程與矩陣運算。課程一開始先介紹多變數函數及函數基底的概念，進而推演一些常用的微分方程，並教導一些數值分析的理論。Topics:

Review of Calculus (Chap. 4, 9, 6, 7)Ordinary / Partial derivativesIntegrationPower Series

Ordinary Differential Equations (Chap. 11, 12, 13)First-order differential equationsSecond and higher order differential equationsSeries solutions of differential equations & special functions

Partial Differential Equations (chap.14)

TEXTBOOK:

The Chemistry Maths Book, Erich Steiner

REFERENCES:

Any Calculus TextbooksAdvanced Engineering Mathematics, E. KreyszigAdvanced Engineering Mathematics, Dennis G. Zill & M. R. CullenMathematical Methods in the Physical Sciences, Mary BoasMathematics for Chemists, C. L. PerrinFoundations of Applied Mathematics, M. D. Greenberg

GRADES: (tentative)

Midterm 30%quiz 30%Final 40%

• Review of Calculus(Chap1.4.6.7.9)• Unit

• Differentiation

• Integration

• Power Series

• Ordinary Differ Equations(D.E)(Chap11-13)• 1st order D.E

• 2nd order and higher D.E

• Series solution of D.E & special functions

• Partial D.E(Chap14)

2

CH1-Numbers,variable,and units

• SI unit• ( System International d’ Unit’s)

• IUPAC• ( International Union of Pure and Applied Chemistry)

3

Atomic Unit(a.u)Table1.2

4

Atomic Unit(a.u)Table1.4

5

Angle SI Unit

2

2

360

90

18'572

360=1rad

6

CH2-Algebraic functions

• ( V , P , T：the dependent variable ; V is a function of the two variable of P , T )

P

nRTV

nRTPV

7

y=2x2-3x+1 quadratic function

y=f(x)= 2x2-3x+1

2a2-3a+1( a: a variable ; a differential operator ; a matrix )

1+)dx

d3(-)

dx

d2(=)

dx

d f(

1+) dx

d3(-)

dx

d2(=)

dx

df(=x

2

2

8

2

3-y=x=(y)f

(y)f find 3+2x=f(x) =y

x=(y)f then f(x)=y If

1-

1-

-1

9

Exanple2.10

• If y=f-(x)=x2+1 express f-1(y)

• y=x2+1 x2=y-1 x=± = f-1(y)

• interchange the x and y axis by rotation around the line x=y

10

11

CH4-Differentiation

• Concept

P

nR

P

V

P

TnRV

P

TTnRVV

TVP

nRTV

nRTPV

,

)(

)(,

12

PP

nRTVV

P

nRTV

TConst

13

• The process of tracking the limlt in Equation is called differentiation

][0

limat xGradient 1

x

y

x

x

xfxxf

xy

x

xdx

dy )()(

0

lim][

0

lim

14

Differentiation from first principle

Operator aldifferenti:D

)()('

dx

dff

dx

dDf

dx

dD

dx

xdfxf

15

微積分基本定理

2

1

)(

1

0

lim

0

lim

)(

1

)(

11)(

1)()(

1)(

)()(

0

lim

0

lim

x

1=yfor principlefirst from Find

xxxxxx

y

xdx

dy

xxxx

y

xxx

x

xxxxfxxfy

xxxfxxfyy

xxfy

x

xfxxf

xx

y

xdx

dy

dx

dD

16

Example4.7

dx

dyey

dx

dyxy

x ,1

,

17

Differentiation by rule (Differentiation from first principle)

18

222

2

3

2235

24632

)12(124*3*

12

)(

)12(12124848

16128)1(2xf(x)y

xxxudx

du

du

dy

dx

dy

xu

uxgy

xxxxdx

dy

xxx

xx

xxy cot

sin

cossinln

19

2

222222

1-

1

cos

1

)22

(

sinsin1cos

cos

1

siny

function tringInverse

xayadx

dy

y

xayaaxaya

yady

dx

dy

dxdx

dy

a

x

求

20

Example4.18

Logarithmic differentiation the appliying

...1

...lnlnlnln

...lnlnlnln

wvuy cba

dx

dw

w

c

dx

dv

v

b

dx

du

u

a

dx

dy

y

dx

wdc

dx

udb

dx

uda

dx

yd

wcvbuay

21

2

3

2

1

2

1

2

2

2

1

2

1

2

1

)1(1

1)

1

1(

1

1

1

1)

1

1

1

1(

2

11

)1

1

1

1(

2

1ln

x)-ln(1-x)(1ln2

1)

1

1ln(

2

1ln

*

,)1(*)1()1

1(y

xxx

x

xdx

dy

xxxdy

dx

y

xxdx

yd

x

xy

vuy

dx

dyxx

x

x

ba

求

)1(ln

,

xxdx

dy

dx

dyxy

x

x 求

22

*Successive differentiation

3

3

3

2

2

2

2

1

ln

xdx

yd

xdx

yd

xdx

dy

xy

23

Stationary point

point saddle 2 x0 2 when x

point minium 3 x03 when x

point maximum 1 x01when x

{ 126

3or 1 when x03139123

point saddle0

min0

max0

min

max { 0

963)(x

saddle

min

max

2

2

2

2

2

232

xdx

d

xxxxdx

dy

dx

yd

dx

dy

xxxxy

y

24

Hϋckel molecular orbital

• C2H4

, 2

1c

, 2

1c

2

1c

02c-1

0c-)c-(1

0(-2c))c-(1*2

1*2c)c-2(1

.)( )c-2c(1)e(

2

22

2

1-

22

1

2

2

1

2-

dc

d

const為，

C

25

Linear and angular motion

Linear

length •arc

locityangular ve 0

lim

t intervalin velocity average

Linear

2

2

dt

d

tt

Angular

dt

xd

dt

dvonaccelerati

dt

dxVvelocity

t

x

26

CH6-Integration

* to find the tangent line to an arbitary curve

→the differential calculus

to find the area enclosed by a given curve

→the integration calculus

CXFXFrulebyationDifferenti

principlefirstfromdx

dyXF

XFy

)()('

)('

)(

27

28

the operation ∫dx is to inverse the effect

of the differentiation

Chap(5).6-Lntegration

Trignomometric Relation

yxyxyx

xxx

xx

xx

x

coscos2

1sinsin

2sincossin

2cos12

1sin

)2cos1(2

1cos

2

2

29

0A2)A1()1(1

)0cos()2cos(0

2cossin

)()()('

xxdxA

aFbFdxxFAb

a

ba

baxx

b

a a

bxxxb

a

x

xx

ee

eeb

ea

e

dxedxedxedxxf

xife

xifeexf

functionsusdiscontinoofnIntegratio

Exapmle

2

)1()1(0

0

)(

0

0 )(

7.5

0

0

30

31

b

c

c

a

b

a

b

a

c

a

b

c

dxxfdxxfdxxf

dxxfdxxfdxxf

)(0

lim)(

0

lim)(

)()(0

lim)(

2)22(0

lim

)1

2(0

lim1

0

lim1

0 1

)(

2

1

2

111

0

xdx

xdx

x

xdefinednotisx

xf

Even and Odd function

f(x)=f(-x) even function偶函數

-f(x)=f(-x) odd function奇函數

32

functionevenxfxfxy ),()(,cos

funevenxfifdxxfdxxfa

a

a

)( ,)(2)(- 0

functionoddxfxfxy ),()(,sin

特殊積分法

1.Substitution method

33

cxcuduu

dxduaxy

dxex

443

3

)12(8

1

8

1

2

1*

2,

1)-(2

令

：

cea

cea

dyea

dya

e

adxdyaxy

dxeex

axyyy

ax

111)

1(

,-

:

令

34

Caa

Cddaa

da

da

a

dadaadaa

daadaaa

daaddxax

dxx

2sin42

-

22cos2

1

22

2cos22

)dcos2-(12

1-

sin)sin(sin)sin(sin

)sin()cos1()sin()cos(

0,sin)cos(,cos

a:ex

22

22

222

2222

2222

22

代入令

35

Caa

Cddaa

da

da

a

dadaadaa

daadaaa

daaddxax

dxx

2sin42

22cos2

1

22

2cos22

)dcos2(12

1

cos)cos(cos)cos(cos

)cos()sin1()cos()sin(

22,cos)sin(,sin

a:ex

22

22

222

2222

2222

22

代入令

36

2.partial fraction method分項積分法

2

7

35

5

35)5()1()1)(5(

)5()1(

.

5ln71ln2

5

7

1

2)

51(

56x

35x2

B

A

BA

BA

xxBxAxx

xBxA

BA

Cxx

dxx

dxx

dxx

B

x

Adx

x

求

求

37

3.Integration by part部分積分法

vduuvudv

vduudvduv

Cxxx

snxdxxxdxxudv

xvdxdu

xdxdvxu

xdxxex

cossin

sincos

sin,

cos,

cos:

令

38

4.Paramteric differentiation method

利用微分求積分

)(

(-n))(-2)(-3)1()()(

)(-2)(-3)1()()(

)(-2)1()()(

)1(1)()(

11)(

:*

......3.2.1,:

10

0

1nn

n

n

0

43

3

3

0

32

2

2

0 0

2

10

0

0

得證！

註

令

前

求

n

axn

ax

ax

ax

axax

aax

axax

nax

a

nex

adxexda

adF

adxexda

adF

adxexda

adF

adxexdxda

de

da

adF

aa

eea

dxeaF

Ca

edxereview

ndxxeex

22

0

00

110

0

1)(

aa

dxa

e)

a

ex)((

vduuvudv

a

edx , vdu

dxe(-x) , dv令u

axax

ax

-ax

註

Chap-7 Sequence and Series

39

)1( , )1

1(

)1(

3210

...:

11

0

lim

0)1

(0

lim

3.2.1 , 1

4.3.2.1 , 1

4

1.

3

1.

2

1.1

9.7.5.3.1

132

1

0

132

32

xx

xaS

xaaxaxSS

axaxaxaxaxxS

axaxaxaxaxaaxS

)... (r

)axaxaxaces(ax of sequenseries:sum

r

r

r

rr

rr

u

sequencesofLimits

rr

u

sequencesHarmonic

sequences

n

n

nn

nn

nn

n

n

r

nnr

n

r

r

r

40

41

42

43

44

45

46

47

48

49

50