1 Mathematics for Chemists, C. L. Perrin Second and higher order differential equations

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 derivatives Integration

Power Series

Ordinary Differential Equations (Chap. 11, 12, 13) First-order differential equations

Second and higher order differential equations

Series solutions of differential equations & special functions Partial Differential Equations (chap.14)

TEXTBOOK:

The Chemistry Maths Book, Erich Steiner REFERENCES:

Any Calculus Textbooks

Advanced Engineering Mathematics, E. Kreyszig

Advanced Engineering Mathematics, Dennis G. Zill & M. R. Cullen Mathematical Methods in the Physical Sciences, Mary Boas

• Review of Calculus(Chap1.4.6.7.9)

• Unit

• Differentiation

• Integration

• Power Series

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

• 1 ^st order D.E

• 2 ^nd order and higher D.E

• Series solution of D.E & special functions

• Partial D.E(Chap14)

CH1-Numbers,variable,and units

• SI unit

• ( System International d’ Unit’s)

• IUPAC

• ( International Union of Pure and Applied Chemistry)

Atomic Unit(a.u)

Table1.2

Atomic Unit(a.u)

Table1.4

Angle SI Unit

2 2

360 90

18' 2 57

= 360 1rad







 





 

CH2-Algebraic functions

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

P V nRT

nRT PV





y=2x ² -3x+1 quadratic function y=f(x)= 2x ² -3x+1

2a ² -3a+1

( a: a variable ; a differential operator ; a matrix )

1 + dx )

3( d -

dx ) 2( d

= dx )

f( d

1 + dx )

3( d -

dx ) 2( d

= dx )

f( d

= x

2

2

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

Exanple2.10

• If y=f ^- (x)=x ² +1 express f ^-1 (y)

• y=x ² +1 x ² =y-1 x=± = f ^-1 (y)

• interchange the x and y axis by rotation

around the line x=y

CH4-Differentiation

• Concept

P nR P

V P

T V nR

P

T T

V nR V

T P V

V nRT

nRT PV

 



 





 









,

) (

)

(

,

P P

V nRT

V     

 P

V nRT

T

Const

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

] 0 [

at x lim

Gradient ₁

x y

x 





 

 



 













 





 

x

x f x

x f x

y x x

dx

dy ( ) ( )

0 ] lim

0 [

lim

Differentiation from first principle

O perator

al differenti

: D

) ) (

( '

dx f df

dx Df d

dx D d

dx x x df

f









微積分基本定理

 

2

1 )

(

1 0

lim 0

lim

) (

1

) (

1 ) 1

(

) 1 ( )

( ) 1 (

) ( )

( 0

lim 0

lim

x

= 1 y for principle

first from

Find

x x

x x x

x y dx x

dy

x x

x x

y

x x

x

x x

x x x

f x

x f y

x x x

f x

x f y

y

x x f y

x

x f x

x f x x

y dx x

dy

dx D d



 







 







 





 











 

 

 













 

















 



 













 

 

 











 



Example4.7

dx e dy

y

dx x dy

y

x , 1

,







Differentiation by rule

(Differentiation from first principle)

2 2

2 2

3

2 2

3 5

2 4

6 3

2

) 1 2

( 12 4

* 3

* 1 2

) (

) 1 2

( 12 12

48 48

1 6

12 8

) 1 (2x

f(x) y









































x x x

dx u du du

dy dx

dy x u

u x

g y

x x

x dx x

dy

x x

x

2

2 2

2 2

2 2

1 -

1 cos

1 2 ) ( 2

sin sin

1 cos

cos 1 sin y

function tring

Inverse

x y a

a dx

dy y

x a

y a

a x

a y

a

y dy a

dx

dy dx dx

dy

a x

 



































求

Example4.18

Logarithmic differentiation the appliying

1 ...

ln ...

ln ln

ln

...

ln ln

ln ln

w v u

y ^{a b c}































dx dw w

c dx dv v b dx du u a dx

dy y

dx w c d

dx u b d

dx u a d

dx y d

w c

v

b

u

a

y

 

  ^{2 1 2 3}

2 1 2

2 2 1 2

1 2

1

) 1

( 1

) 1 1

( 1 1

1

1 ) 1 1

1 1

( 1 2 1 1

1 ) 1 1

( 1 2 1 ln

x) - ln(1 - x) (1 2 ln

) 1 1

ln( 1 2 ln 1

*

, ) 1

(

* ) 1

( 1 )

( 1 y

x x x

x x

dx dy

x x

x dy

dx y

x x

dx y d

x y x

v u

y

dx x dy

x x x

b a





 



 

 

 

 

 

 



 

 







 

  ^ 求

 , dy x

y ^x 求

*Successive differentiation

2 2

2

1 ln

 







dx x y d

x dx

dy

x

y

Stationary point

  

point saddle

2 x

0 2 when x

point minium

3 x

0 3 when x

point maximum

1 x

0 1 when x {

12 6

3 or 1 when x 0

3 1

3 9 12

3

point saddle

0

min 0

max 0

min { max 0

9 6

3) (x

min max 2

2

2 2

2

2 3

2

































































dx x d

x x

x dx x

dy dx

y d dx dy

x x

x x

y

y

Hϋckel molecular orbital

• C ₂ H ₄



 































2 c 1

0 2c

- 1

0 c

- ) c - (1

0 (-2c)

) c - (1 2 *

* 1 2c )

c - 2(1

.) (

)

c - 2c(1 )

e (

2

2 2

2 -1 2 2

1 2

2 1 2 -

dc d

const 為

，

C

Linear and angular motion

Linear

locity angular ve

lim

t interval in

velocity average

Linear

2 2

 

 _{ }













 



d Angular

dt x d dt

on dv accelerati

dt V dx

velocity t

x

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

principle first

dx from X dy

F

X F y





)

( '

)

(

the operation ∫dx is to inverse the effect

of the differentiation

Chap(5).6-Lntegration

Trignomometric Relation

 

   

 ^{x y x y} 

y x

x x

x

x x

x x

x



















cos 2 cos

sin 1 sin

2 sin cos

sin

2 cos 2 1

sin 1

) 2 cos 1

2 ( cos 1

2 2

 

0 A2) A1

( ) 1 ( 1

) 0 cos (

) 2 cos 0 (

cos 2 sin

) ( )

( )

( '







































  x xdx

A

a F b

F dx x F

A ^b

a

b a

x x

b

a a

b x

x x b

a

x x x

e b e

e e

dx e

dx e

dx e

dx x

f

x if

e

x if

e e x

f

functions us

discontino of

n Integratio Exapmle







 

 



 















 





 



  



) 1

( )

1 0 (

) (

0

0

) (

7 . 5

0

0

 







  









 

 



 



b c c

a b

a b a

c a

b c

dx x f dx

x f dx

x f

dx x f dx

x f dx

x f















) 0 (

) lim 0 (

) lim (

) ( )

0 ( ) lim

(

1 1 lim

1 lim

0

1 )

(

1 1 1









x defined not

x is x

f

Even and Odd function

f(x)=f(-x) even function偶函數 -f(x)=f(-x) odd function奇函數

function even

x f

x f

x

y  cos , ( )  (  ), fun even

x f if dx x f dx

x

a f

a

a ( ) , ( ) 2

)

- ( 0

 ^ 

function odd

x f x

f x

y  sin , (  )  ( ),

特殊積分法

1.Substitution method

c x

c u

du u

dx du

ax y

dx ex

























4 4

3

3

) 1 2

8 ( 1 8

1 2

* 1

2 ,

1) - (2 令

：

 ^{e  dx}

ex : ^ax

a C a

C d

a d a

a d a d

a

d a

d a

a d

a a

d a

a d

a a

a

d a

a d dx a

x

dx x







































































  











































































2 4 sin

- 2

2 2

2 cos 1 2

2

2 2 cos

)d 2 cos2 -

2 (1 - 1

sin )

sin (

sin )

sin (

sin

) sin (

) cos 1

( )

sin (

) cos (

0 , sin

) cos (

, cos a :

ex

2 2

2 2

2 2

2

2 2

2 2

2 2

2 2

2 2

代入

令

C d

a d a

a d a d

a

d a

d a

a d

a a

d a

a d

a a

a

d a

a d dx

a x

dx x





















































  



























































 

 







2 2

1 cos

2 2 cos

)d 2 cos2

2 (1 1

cos )

cos (

cos )

cos (

cos

) cos (

) sin

1 ( )

cos (

) sin (

2 , 2

cos )

sin (

, sin a :

ex

2 2

2 2

2

2 2

2 2

2 2

2 2

2 2

代入

令

2.partial fraction method分項積分法

 







 

 























 























 



 

 

 





   



2 7 3

5

5

3 5

) 5 (

) 1 ) (

1 )(

5 (

) 5 (

) 1 (

.

5 ln

7 1

ln 2

5 7 1

) 2 5 ( 1

5 6

x

3 5x

2

B A B

A

B A

x x

B x

x A x

x B x

A

B A

C x

x

x dx x dx

x dx B x

dx A x

求

求

3.Integration by part部分積分法

 ^{ } 





vdu uv

udv

vdu udv

duv

x v

dx du

xdx dv

x u

xdx x

ex











sin ,

cos ,

cos :

令

4.Paramteric differentiation method 利用微分求積分

)(-2)(-3) 1

( )

) ( (

)(-2) 1

( )

) ( (

) 1 (

1 )

) ( (

1 ) 1

(

:

*

...

3 . 2 . 1 ,

:

0

4 3

3 3

0

3 2

2 2

0 0

2 1 0

0 0

註 令

前 求

  

  

    





  

 

 







































 









 







ax ax

ax ax

a ax

ax ax

n ax

a dx

e da x

a dF

a dx

e da x

a dF

a dx

e x da dx

de da

a dF

a a e

a e dx

e a

F

a C dx e

e review

n dx x

e ex





2 2

0 0

0

1 0 1

0 1)

(

a a

a dx ) e

a x)( e (

vdu uv

udv

a dx , v e

du

dx e

(-x) , dv 令u

ax ax

ax -ax









 

 







 















 









註

Chap-7 Sequence and Series

3 2 1 0

. .

. : 1 1

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

1 2 3 1

3 2



















 



 









 

 ^{ax a ax ax ax ax ax}

S

) .

. . (r

) ax

ax ax a ces(ax

of sequen series:sum

r r r

r r

r r u

sequences of

Limits

r r u

sequences Harmonic

sequences

n r n n

r r

r



















