The Mathematics for Chemists (I)

(Fall Term, 2004)(Fall Term, 2005)(Fall Term, 2006)

Department of ChemistryNational Sun Yat-sen University

化學數學（一）

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.92:74p.122: 32pp. 142-144: 18,37,45,51,66,71pp.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 numberwas thrown into sea.

Units (base)meter (m) distance "The metre is the length of the path travelled by light in vacuum during

a time interval of 1/299 792 458 of a second."

kilogram (kg)

mass "The kilogram is equal to the mass of the international prototype of the kilogram."

second (s)

time "The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom."

ampere (A)

electric current "The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 metre apart in vacuum, would produce between these conductors a force equal to 2 × 10-7 newton per metre of length."

kelvin (K) temperature "The kelvin is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water."

mole (mol)

amount of substance

"The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12. When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles."

candela (cd)

intensity of light "The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 × 1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian."

Derived Unit Measures Derivation Formal Definition

hertz (Hz) frequency /s s-1

newton (N) force kg·(m/s2) kg·m·s-2

pascal (Pa) pressure N/m2 kg·m-1·s-2

joule (J) energy or work N·m kg·m2·s-2

watt (W) power J/s kg·m2·s-3

coulomb (C) electric charge A·s A·svolt (V) electric potential W/A kg·m2·s-3·A-1

farad (F) electric capacitance C/V kg-1·m-2·s4·A2

ohm (Ω) electric resistance V/A kg·m2·s-3·A-2

siemens (S) electric conductance A/V kg-1·m-2·s3·A2

weber (Wb) magnetic flux V·s kg·m2·s-2·A-1

tesla (T) magnetic flux density Wb/m2 kg·s-2·A-1

henry (H) inductance Wb/A kg·m2·s-2·A-2

degree Celsius (°C) temperature K - 273.15 K

radian (rad) plane angle m·m-1

steradian (sr) solid angle m2·m-2

lumen (lm) luminous flux cd/sr cd·sr-1

lux (lx) illuminance lm/m2 m-2·cd·sr-1

becquerel (Bq) activity /s s-1

gray (Gy) absorbed dose J/kg m2·s-2

sievert (Sv) dose equivalent Gy·(multiplier) m2·s-2

katal (kat) catalytic activity mol/s mol·s-1

Units (derived)

Metric Prefixes

yotta- (Y-) 1024 1 septillion

zetta- (Z-)1021 1 sextillion

exa- (E-) 1018 1 quintillionpeta- (P-) 1015 1 quadrilliontera- (T-) 1012 1 trilliongiga- (G-) 109 1 billionmega- (M-) 106 1 millionkilo- (k-) 103 1 thousandhecto- (h-) 102 1 hundreddeka- (da-)** 10 1 tendeci- (d-) 10-1 1 tenthcenti- (c-) 10-2 1 hundredthmilli- (m-) 10-3 1 thousandthmicro- (µ-) 10-6 1 millionthnano- (n-) 10-9 1 billionthpico- (p-) 10-12 1 trillionthfemto- (f-) 10-15 1 quadrillionthatto- (a-) 10-18 1 quintillionthzepto- (z-) 10-21 1 sextillionthyocto- (y-) 10-24 1 septillionth

Table: Basic quantities for the atomic unit system

Constant Symbol

rest mass of the electron me

elementary charge e

Planck's constant divided by               

    times the permittivity of free space      

Atomic Units

Table: Quantities for the atomic unit system

Constant Symbol Recommended value

length, Bohr                     5.291 772 49(24)          m

velocity,           2.187 691 42(10)        ms-1

energy, Hartree                 4.359 748 2(26)          J

time             2.418 884 326 555(53)          s

magnetic dipole moment                9.274 015 4(31)          JT-1

electric dipole moment d0= ea0 8.478 357 9(26)          Cm

Variables, Algebra and Functions

• Domain (of definition, of value) tvs *,...3,2,1,222

02

4

32 nE

n

emn

e

• Continuous vs discrete

a+b=b+a, ab=ba (commutative)a+(b+c)=(a+b)+c, (ab)c=a(bc), (associative)a(b+c)=ab+ac (distributive)

uuupqqpyxy baabaaaa )(,)(,a

lawIndex x

Polynomials

nn xaxaxaaxf ...)( 2

210

n

i

ii xaxf

0

)(

))...()((

...)(

21

2210

nn

nn

xxxxxxa

xaxaxaaxf

Roots (zeros of f(x))

Factorization:

Rational Functionsn

n xaxaxaaxP ...)( 2210

nn xbxbxbbxQ ...)( 2

210

nn

nn

xaxaxaa

xbxbxbbxPxQxf

...

...)()(

2210

2210)(

Singularity(奇點 ):

x

f

x0

0|)( xxxf

(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

xxxxxf

xxf

aexf

xxxxxf

xxxxxf

n

xx

ctanh,tanh,cosh,sinh)(

log)(

,)(

actan,atan,acos,asin)(

ctan,tan,cos,sin)(

xx

eeeeee

nnnxnn

yxyx

xx

x

xxxxxx

yxyxyx

xyyxyx

xx

xxx

yxxyx

yx

yxyxyxxyyxyx

xx

xxecxx

xxxxxx

x

xxxxx

11

211

2121

22

11

22

1

tantan1tantan

22

sin1

cos1

lntanh

]1ln[sinh],1ln[sinh

sinhsinhcoshcosh)cosh(

coshsinhcoshsinh)sinh(

1sinhcosh

tanh,cosh,sinh

loglog)(log,loglog

)tan(

sinsincoscos)cos(,cossincossin)sin(

1cossin

cot)cot(,cos,sec

tan)tan(,cos)cos(,sin)sin(

2

2

Classroom Exercise

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

xx 2log),4sin(

Complex Functionsiyxzzf ),(

iccci ebrabzazfrezzf

ryrx

)(,)(

sin,cos

)]...isin()...[cos(...rrr...zzz

1,2,...ni),isin(cosrz

:formula sMoivre' de

n21n21n21n21

iiii

x

y

rθ

(Proof) (classroom exercise)

1 2

1 2

1 2

1 1 2 2

1 2 1 2

( ... )1 2

1 2 1 2 1 2

cos , sin

, ...

... ... ...

...

... [cos( ... ) sin( ... )]

n

n

n

i

ii in n

ii in n

in

n n n

x z re r y r

z re z r e z r e

z z z r r r e e e

r r r e

r r r i

Proof (by Mathematical Induction)

)]...isin()...[cos(...rrr...zzz

1,2,...ni),isin(cosrz

:formula sMoivre' de

n21n21n21n21

iiii

)]sin()[cos(rr

)sinsin-sincososcsincos(cosrr

)sin)(cossin(cosrrzz

2nWhen

212121

2121212121

22112121

i

ii

ii

)]sin()[cos(...rrr...zz

:knfor holdseqution that theSuppose

2121k2121 kkk iz

1 2 k k 1 1 2 k 1 2 k 1 2 k k 1 k k 1

1 2 k k 1 1 2 k k 1 1 2 k k 1 1 2 k k 1 1 2 k

With n k 1

z z ...z z r r ...r [cos(θ θ ... θ ) isin(θ θ ... θ )]r (cosθ isinθ )

r r ...r r [cos(θ θ ... θ )cosθ isin(θ θ ... θ )cosθ cos(θ θ ... θ )sinθ -sin(θ θ ... θ )i

k 1

1 2 k k 1 1 2 k k 1 1 2 k k 1

sinθ ]

r r ...r r [cos(θ θ ...θ θ ) isin(θ θ ... θ θ )]

Recall the properties oftrigonometric 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

)!(!!

0

1222

)1(222

)1(1

,

......)(

knknk

n

n

k

kknkn

nnnnnkknkn

nnnnnn

k

nCyxC

ynxyyxyxCyxynxxyx

n

k

kkn

n

k

kknkn

nnnnnkknkn

nnnnnn

yCyC

yynyyCyyny

00

1222

)1(222

)1(1

1

11...1...111)1(

11

11 ...1)...1)(1(1 n

yynn yyyyyy

n

)1(...21 21

212

21

1

nnnnnrn

r

)12)(1(...21 61

612

213

31222

1

2

nnnnnnnrn

r

22412

413

214

41333

1

3 )1(...21

nnnnnnrn

r

nnnnnrn

r3013

314

215

51444

1

4 ...21

)2)(1()1( 31

1

nnnrrn

r

)3)(2)(1()2)(1( 41

1

nnnnrrrn

r

11

)1(1

nn

n

rrr

)2)(1(21

41

1)2)(1(

1

nn

n

rrrr

Classroom exercise: Prove any of above sums

Important Infinite Series...1 !3!2

0!

32

xx

kkxx xe

k

...1 51

31

012

)1(4

kk

k

...sin !5!30

)!12()1( 5312

xx

kk

x xxkk

Classroom exercise: Prove

11 ...,)1ln( 432

432 xxx xxx

Convergence and Divergence...1 5

131

012

)1(4

kk

k

...1 31

21

1

1

kk

...3211

aaaar

r

Necessary for convergence: r as 0ra

Further test of convergence:

By comparison:0...,321

1

r

rr aaaaa 0...,321

1

r

rr bbbbb

d’Alembert’s ratio test:

1rra

ateindetermin 1|| if

divergent 1|| if

convergent 1|| if

1

1

1

limlimlim

r

r

r

r

r

r

aa

r

aa

r

aa

r

...1 31

21

1

1

kk

(unbelievable billionaire!)

Limit as the Core of Modern Mathematics

).()(lim

: is whenoflimit thecall then we,|)()(|

, that whenso an exits thereis,it small howmatter no ,any for If

0xx

000

0

0

xfxf

)f(xx x f(x)xfxf

||x-x

4lim24

2

2

x

x

x

?|)( 2242

xxxxf

x 2.10 2.01 2.001 2.0001 … 1.9999 1.999 1.99 1.9

y 4.1 4.01 4.001 4.0001 … 3.9999 3.999 3.99 3.9

Find the Limit of a Function

,,

:limit of classes typicalThree

00

4)2(lim|)(2

2242

xxf

xxx

x

10)105(lim})3()2{(lim 2

0

2121

0

xxx

xxxx

2)(lim)(lim/31/52

352 2

2

2

x

x

xxxx

x

lim [ln(2 3) ln( 2)] ln 2x

x x

(Classroom exercise)

Differentiation as Limit of Division

}{limlim )()(

00 xxfxxf

xxy

xdxdy

xx

exxe

xxee

xxy

xdxdy

x

e

eyxx

xxx

}{lim}{limlim

...)1(

000

221

x

y=f(x)

2

)(11

1

000

1

}{lim}{limlimxxxxxx

y

xdxdy

x

xxxx

xxx

y

?, dxdyxy

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

x

xx

aa

x

x

xxx

ee

xx

xx

xx

axx

const

DerivativeFunction

1

2

1

cosh

ln

sinhsinhcosh

sectan

sincos

cossin

0

Common Rules for Differentiation

dx

dy

dxdu

dudf

dxdv

dxdu

vu

dxd

dxdu

dxdv

dxd

dxdv

dxdu

dxd

dxdu

dxd

uf

vuv

vuuv

vu

aau

RuleType

1dydx

dxd

2

rule inverse

))((rulechain

/)()(quotient

)(product

)(sum

)(multiple

Frequently Used Derivatives

2 2

2 2

2 2

1 1

1 1

1

(sin )

(cos )

(tan )

xa a x

xa a x

x aa a x

D

D

D

2 2

2 2

2 2

1 1

1 1

1

(sinh )

(cosh )

(tanh )

xa x a

xa a x

x aa a x

D

D

D

dxdD

:Operator alDifferenti

Implicit Function

?,0),( Dyyxfdxdy

251

45

5

4

0125)()2()(

02),(

yDy

DyDyyxDyDyDDf

xyyyxf

Successive Differentiation

,...''','','

)(32 fDyfDyDfy

xfy

even) is (if sin)1()(

...

)(sin)(''

cos)('

sin)(

2)(

22

naxaxf

xfaaxaxf

axaxf

axxf

nnn

odd) is (if sin)1()( 21

)( naxaxf nnn

How about odd n? (Classroom exercise)

Stationary Points

reflection ofpoint 0,0

minimum 0,0

maximum 0,0

point stationary 0

2

2

2

2

2

2

dx

yddxdy

dx

yddxdy

dx

yddxdy

dxdy

A,B,C

A

C

B

Turning points

A

B

C

A

B

C

D

E

F

Local minima: E,C Global minimum: C

Local maxima: A, D, FGlobal maximum: D

Snell’s Law of Refraction

P

Q

Phase boundary

θ2

θ1r1

r2

O

y2

x1x2

y1

constant. are and 2,121 yyxxX

2

2

1

1

vr

vrt

2/121

21

2/122

222

2/121

211 ))(()(,)( yxXyxryxr

01

dxdt

To find point O so that the time used forthe light to travel from P to Q is minimized.(Principle of least time)

2

1

2

1

2

2

1

1

2

12

21

1

12

2

21

1

11

sinsin

sinsin1111 0

vv

vvr

x

vrx

vdxdr

vdxdr

vdxdt

2

1

1

2

2

1

2

1

sinsin

//

nn

ncnc

vv

Maxwell-Boltzmann Distribution of Speed

RTMv

evRT

Mvf 22

23

2

24

v*

21

)(

2

0| *

M

RTv

vvdvvdf

(Classroom exercise:Verify the expression for the most probable speed.)

Consecutive elementary reactions

Akdt

Ada

IkAkdt

Idba

Ikdt

Pdb

tkaeAA 0

tkab

aeAkIkdt

Id 0

0Aeekk

kI tktk

ab

a ba

01 Akk

ekekP

ab

tkb

tka

ab

PIA ba kk PuNpU239day 2.35239min 2.35239

Classroom exercise:Find the maximum of species I.

MacLaurin Series

0

44

33

2210 ...)(

k

kk xcxcxcxcxccxf

...

...234232)('''

...34232)(''

...432)('

432

2432

34

2321

xcxccxf

xcxccxf

xcxcxccxf

nn cnf

cfcfcfcf

!)0(

,...!3)0(''',!2)0('',)0(',)0()(

3210

2 3 ( )1! 2! 3! !

0

( ) (0) '(0) ''(0) '''(0) ... (0)k nx x x x

kk

f x f f f f f

Taylor Series

0

44

33

2210 ...)(

k

kk xcxcxcxcxccaxf

...

...234232)('''

...34232)(''

...432)('

432

2432

34

2321

xcxccaxf

xcxccaxf

xcxcxccaxf

nn cnaf

cafcafcafcaf

!)(

,...!3)(''',!2)('',)(',)()(

3210

0

)(!)(

!3)(

!2)(

!1)(...)(''')('')(')()(

32

k

nkaxaxaxax afafafafafxf

k

0

)(!!3!2!1

)(...)(''')('')(')()(32

k

nkxxxx afafafafafaxf

k

Approximation of Series2 3

1

( ) ( ) ( ) ( )1! 2! 3! !

( ) ( 1)( 1)!

( ) ( ) '( ) ''( ) '''( ) ... ( ) ( )

( ) ( )

k

n

x a x a x a nx ank

x a nn n

f x f a f a f a f a f a R x

R x f b

Taylor’s theorem:

xbexR

xRxe

bnx

n

nnxxxx

n

n

0,)(

)(...1

)!1(

!!3!21

32

xnx

nxxxx

nx

nxxx

xnx

nnx

exex

exR

nnnn

nn

)!1(!!3!2)!1(!!3!2

)!1()!1(

132132

11

...1...1

)(

?lim

0)()(

)()(

xgxf

ax

agaf

...)(''')('')(')()(!3

)(!2

)(!1

32

afafafafxf axaxax

...)(''')('')(')()( !3)(

!2)(

!1

32

agagagagxg axaxax

)(')('

)(')('

)('0

)('0

...)(''')('')(')(

...)(''')('')(')(

)()(

limlim

limlim

0)()(

!1

!1

!3

3)(

!2

2)(

!1

!3

3)(

!2

2)(

!1

xgxf

axagaf

ag

af

ax

agagagag

afafafaf

axxgxf

ax

ax

ax

axaxax

axaxax

agaf

l’Hôpital’s Rule

Approximation of Series

x

xxx

x

xxx

small

43211...,)1ln(

432

xe

xx

xx

xx

x

x

1

1cos

sin

small

221small

small

Integration as Limit of Sum

dxxfxxfb

a

N

ii

N

)()(lim

1

Common Rules for Integration

dxdF(x)b

a

N

ii

Nf(x)aFbFdxxfxxf

with )()()()(lim1

The fundamental theorem of the calculus:

The definite and indefinite integrals.

dxvuvdxu

dxxvdxxudxxvxu

dxxuadxxau

dxdu

dxdv

)()())()((

)()(

a

b

b

a

b

e

e

d

d

c

c

a

b

c

c

a

b

a

b

a

b

a

b

a

dxxfdxxf

dxxfdxxfdxxfdxxfdxxfdxxfdxxf

duufdttfdxxf

)()(

)()()()()()()(

...)()()(

Elementary Integrals

x

xx

aa

x

x

xxx

ee

xx

xx

xx

axx

const

DerivativeFunction

1

2

1

cosh

ln

sinhsinhcosh

sectan

sincos

cossin

0

Cxa

Cax

xa

ax

Caxax

Cee

Caxax

Caxax

Caxax

aCaxx

Caxa

IntegralFunction

a

a

axa

axa

a

a

aa

ln

cosh

/

sinh

sinhcosh

tansec

sincos

cossin

)1( )1/(

1

1

1

12

1

1

1

Average of a Function

b

a

b

a

dx

dxxf

abAy

)(_

2

0/ 2

0

sinsin ?

d

dx

ba

x

y=f

x

y

Integration of Odd/Even Functions

aa

a

dxxfdxxf

xfxf

0

)(2)(

)()(

0)(

)()(

a

a

dxxf

xfxf

Special Case: Discontinuous Functions

3

2

2

1

3

1

9)12(2)(

2 if 12

2 if 2

dxxxdxdxxf

xx

xxf(x)

2

z

y

n

c

b

b

a

z

a

n

dxxfdxxfdxxfdxxf

xf

xf

xfxf

f(x)

)(...)()()(

zy if (x)

...

dc if (x)

cb if (x)bxa if )(

21

3

2

1

Special Case: Improper Integrals

ca c+ε bc-ε

})()({lim)(0

b

c

c

a

b

a

dxxfdxxfdxxf

2)22lim(|2lim0

11

1

0

1

0

1

xdxdxxx

)1lim(|}{lim 111

1111

11

0

1

0

1 aaa axax

dx

Special Case: Infinite Integrals

b

ab

a

dxxfdxxf )(lim)(

aab

b

ba

r

b

b

a

r

ba

r eeeedredre

][lim][limlim

ExampleCalculate the mean speed of N2 at 25 oC

21

22212/3

2

0

2/32/32

0

8

)()(4

)(4)(2

M

RT

dvevdvvvfc

MRT

RTM

RTMvRT

M

2

00

0 0

22

02

13

2/10

2/][

2/2/2

2

2

a

dyedyey

dyyedxex

dxex

ayay

ayax

a

ax

m/s 475)( 2/1

kgmol0102.28298Kmol8.3141JK8

13

11

c

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.

222

2

1

2

1

2

1zyxK mvmvmvE

02

3

][

][)(:energy internalMolar

23

21

21

121

2212

21

1

221

m

PA

PN

i

Pzy

N

i

xm

URT

UkTNUkTkTkT

UmvmvmvTU

A

A

(Monatomic gas)

Functions of Several Variables

pnRTnTpfV ),,(

ji

iji

mv rUE i )(2

2

22 32),( yxyxyxf

x

y

z

z

yx

P

O

22 32),( yxyxyxfz

Partial Differentiation

}{limlim

}{limlim

),(),(

00

),(),(

00

yyxfyyxf

yyz

yyz

xyxfyxxf

xxz

xxz

yx

yx

yxyxyxfz

yz

xz

62

22

32),( 22

x

y

z

z

yx

P

O

Constant volume

Vv T

UC

),( VTUU

The Relation Between Cp and Cv

pp T

HC

VV

UC

T

nRCC Vp (perfect gas)

( ) ( )H Up V p VT TC C

Common Rules for Partial Differentiation

x

y

xu

uf

dxdv

dxdu

vu

x

xu

xv

x

xv

xu

x

xu

x

uf

vuv

vuuv

vu

aau

RuleType

1yx

x

2

rule inverse

))((rulechain

/)()(quotient

)(product

)(sum

)(multiple

Change of Variables(Coordinate Transform)

sin,cos sysx

sincosyz

xz

sy

yz

sx

xz

sz

cossinyz

xz

yyzx

xzz

ss

vy

yz

vx

xz

vz

uy

yz

ux

xz

uz

vugyvufx

),(),,(

sθx

y

VnRTTVfp ),(

Vn

TVp

)(

mV

mp

Vn

qVp

C

C

,

,0 ,)(

Higher Derivatives

2222

3223

1262,343

432),(

yxyxyxyx

yxyyxxyxfz

yz

xz

yxyx

yxyx

xyz

yz

xyz

yz

y

xyz

xz

yxz

xz

x

64)(,246)(

64)(,46)(

2

2

2

2

2

2

yxz

xyz

22

,...,,,32

2

2

zyxf

xyzxyf

xyx

fxxx

fx ffff

Higher Derivatives

VpTnG

mG ,,)(

),,,( nVTpGG

dVV

UdS

S

UdU

sv

TS

U

V

p

V

U

S

pdVTdSdU

sV

USvS

UV

VS Sp

VT )()(

Total Differential

x

y

z

z

ΔyΔx

P

O

QR

p

rq

dydxdz

yyxxyxz

yxfz

xyz

yxz

xyz

yxz

yxz

xyz

yxz

)()(

...)()()()()()(

),(2

212

21

2

22

2

2

n

iiijxx

zxxx

zxxx

z

n

dxdxdxdz

xxxxfz

ji1

},{2,1,

321

)(...)()(

),...,,(

,...312,...321

dTT

UdV

V

UUU'

VT

dTT

UdV

V

UdU

VT

Change of Internal Energy as a Total Differential

Tpm

nTV

npV

m

TpnTnp

n

VV

p

V

T

V

dnVVdpVdT

dnn

Vdp

p

VdT

T

VdV

nTpVV

,,

1

,

1

,,,

,,

),,(

Volume as a Total Differential

The Total Derivative

n

idtdx

ijxxz

dtdx

xxxz

dtdx

xxxz

dtdz

iin

i

ji

txxxxxxfz

1},{,,

321

)(...)()(

)(),,...,,(

2

,...312

1

,...321

??,

,,),( 133

dtdz

dxdz

tx

exyyxyxfz

Classroom exercise:

The -1 Rule (Chain Rule)

zyx

yzx zx

yyx

z

zxy

xyz

yxz

xyz

yxz dydxdz

)(1

)(1 )( ,)( Using

0)()()(

)()(0

0?dz ifabout How

)()(

),(

dydxdz

yxfz

xyz

yxz

1)()()(

xzy

zyx

yxz

Using The -1 Rule

Tpm

nTV

npV

TpV

VpT

Tp

n

VV

p

V

T

V

p

V

T

V

T

p

T

p

V

T

p

V

dpp

VdT

T

VdV

nnTpVV

,,

1

,

1 ,,

/

1

fluid. ain fixed with ),,(

Differential with Constraint

uxy

xyz

yxz

uxy

xyz

uxx

yxz

uxz

xyz

yxz dydxdz

constyxguyxfz

)()()()()()()()(

)()(

),(),,(

xyz

yx

yxz

uxz

yx

xyu

yxu

uxy

uxy

xyz

yxz

uxz

constayxuyxfz

)()()(

)/()()(

)()()()(

),,( 222

Example:

, , , =const

p V T p

p p p

p p V T p

p Vp V

U U V T p p V T

U U U V

T T V T

U H VU H pV p

T T T

H V U U Vp

T T T V T

H UC C

T T

T p

U Vp

V T

Prove that the difference of the isobaric and isochoric heat capcities is

= , = ,p V p Vp V T p p

H U U V VC C C C p

T T V T T

Example

Change of Independent Variables(Coordinate Transform)

uvy

xyz

uvx

yxz

uvz

vuy

xyz

vux

yxz

vuz

uvz

vuz

xyz

yxz

dvdudz

dydxdz

vuyyvuxxyxfz

)()()()()(

)()()()()(

)()(

)()(

),(),,(),,(

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

z z u z vy v y u yx u x v x

z z u z vx v x u xy u y v y

Classroom Exercise

?)(?,)(

?)(?,)(

sin,cos),,(

xyz

yxz

rz

rz

ryrxyxfz

xyz

yxz

xyz

yxz

ry

xyz

rx

yxz

rz

yx

yxy

z

yx

xyx

zxy

zyx

zry

xyz

rx

yxz

rz

xyrr )()()(cos)(sin)()()()()(

)()(sin)(cos)()()()()()(2222

rz

rrz

xyrz

xyr

rz

xyz

rz

rrz

yxrz

yxr

rz

yxz

)(sin)()()()()()(

)(cos)()()()()()(

cos

sin

Exact Differentials

xyz

yxz GF

GdyFdx

yxGGyxFF

)(,)(

),(),,(

yxG

xyF

GdyFdxdz

)()(

yxG

xyF

xydydxyxGdyFdx

)()(

2)( 22

yxG

xyF b

dycybxdxbyaxGdyFdx

)()(

)2()2(

Stationary Points

x

y

z

s

M1

M2

m1

m2

M3

M4

point saddle000000

point saddle000000

maximum000000

minimum000000

2

Natureffffffff xyyyxxxyyyxxyx

Stationary Points

point saddle000000

point saddle000000

maximum000000

minimum000000

2

Natureffffffff xyyyxxxyyyxxyx

point saddle2241684

point saddle2241684

maximum2520241202

minimum2520241202

),(),,(),0,2(),0,2(:points stationary

612,1263

1226),(

34

32

34

32

2

34

32

34

32

222

323

Natureffffffyx

yxyfyxf

xyxyxyxf

xyyyxxxyyyxx

yx

Optimization with Constraints(Method of Lagrange Multipliers)

constzyxg

zyxf

),,(:constraintwith

),,( of valueextremum theFind

0,0,0

:sMultiplier Lagrange of method The

zg

zf

yg

yf

xg

xf

miaxxxg

xxxf

ini

n

...3,2,1, ),....,(:constraintwith

),....,( of valueextremum theFind

21

21

miaxxxg

nk

gfgggf

ini

m

ixg

ixf

x

m

iiimm

k

i

kk

...3,2,1, ),....,(

,...3,2,1,0

...

:sMultiplier Lagrange of method The

21

1

12211

Optimization with Constraints(Method of Lagrange Multipliers)

constxzyzxyA

xyzV

)(2:constraintwith

:area surfacegiven afor lumelargest vo ofbox r rectangula a of dimensions theFind

Axyyzxzg

zyxkkkk x

gxf

x

)(2

),,(3,2,1,0

Axyyzxzx

yzxzxyAx

xyzx

xyzxzxyA

xxyz

xxyzxzxyA

xxyz

x

)(2,0

,0,0

)](2[)(

)](2[)()](2[)(

23

)(6AVzyx

Secular Equation

.1 ),....,(:constraintwith

, ),....,(

of valuesstationary theFind

1

221

1 121

n

iin

jiij

n

i

n

jjiijn

xxxxg

CCxxCxxxf

2

2232232222

11311321122111

2332211

22322322221221

11311321122111

1

2

1 1

)(

...

2...2)(

2...22)(

)(...2

......)(

...)(

:smultiplier Lagrange of method theUsing

nnn

nn

nn

nnnnnnnnn

nn

nn

n

ii

n

i

n

jjiij

xC

xxCxxCxC

xxCxxCxxCxC

xCxxCxxCxxC

xxCxxCxCxxC

xxCxxCxxCxC

xxxCgf

0)(2...222

...

02...2)(22

02...22)(2

0

332211

2323222121

1313212111

nnnnnn

nn

nn

x

xCxCxCxC

xCxCxCxC

xCxCxCxC

k

njxCn

iiijij ,...3,2,1,0)(

1

Curvilinear Integrals

CC

dsyxfdmM ),(

x

y

A

BC

ds dy

dx

a x x+dx b

dxdsdxdy 2/12 ])(1[

C

dxdy dxxyxfI 2/12 ])(1))[(,(

b

adxdy

C

dxyxGyxFdyyxGdxyxFI ]),(),([]),(),([

0

131

422

1

2

]1[)(

1

,

2

dxxxxydyydxI

dxdxdy

xy

xyGyF

C

yx

x

x

B

A

1

1

0x

y

Curvilinear Integrals Independent of Path

C

dyyxGdxyxFI ]),(),([

dzGdyFdx

GF xyz

yxz

)(,)(

),(),(]),(),([ 1122 yxzyxzdzdyyxGdxyxFICC

It depends on the initial and final coordinates only.

B

A

1

0

x

y

(x1,y1)

(x2,y2)

Entropy: A State Function

dpp

SdT

T

SdS

nnTpSS

Tp

fluid. ain fixed with ),,(

pp

pV

T

HC

T

V

1

2

1

1

0

T

p

(p1,T1)

(p2,T2)

C Tp

dpp

SdT

T

S

TpSTpSS ),(),( 112212

Which path is easier for us to calculate?

2

1

2

1 1

2

1

2

1 11

),(),(

),(),(

222123

112131

233121

p

p

p

p TT

T

TT

CT

T ppC

Vdpdpp

S

TpSTpSS

dTdTT

Sds

TpSTpSS

SSS

p

Multiple Integrals

f

e

d

c

b

a

d

c

b

a

d

c

b

a

dzdydxzyxf

dxdyyxfdydxyxf

}]),,([{

}),({}),({

2

x

y

(0,0)

(1,1)

R y=x2

y=x

RdxdyyxfI

xyf(x,y)

),(

R range over the 21 of integral theCalculate

1

0

})21({2

dxdyxyIx

x

41

1

0

1

0

5322 ][}]{[ 2 dxxxxxdxxyyI xx

41

1

0

1

0

322

1

0

][}]{[

})21({

dyyyyydyyxx

dydxxyI

yy

y

y

yxxy

yxxy

2

Change of Variables

wz

vz

uz

wy

vy

uy

wx

vx

ux

f

e

d

c

b

a

f

e

d

c

b

a

J

dudvdwJwvuzwvuywvuxfdxdydzzyxf

wvuzzwvuyywvuxx

'

'

'

'

'

'

||)),,(),,,(),,,((),,(

),,(),,,(),,,(

vy

uy

vx

ux

d

c

b

a

d

c

b

a

J

dudvJvuyvuxfdxdyyxf

vuyyvuxx

' '

'

||)),(),,((),(

),(),,(

( , )yxv vdv dv

( , )yxu udu du

area of R(x,y)

( , ,0) ( , ,0)

0

0

| |

| ( , , , ) |

y yx xu u v v

yxu u

yxv v

y yx xu v v u

d

du du dv dv

i j k

du du

dv dv

dudv

J x y u v dudv

A dB

dAdB

( , )yxv vx dv y dv

( , )yxu ux du y du

2D proof of integration with changed variables

(x,y)

?0

2

dxeI x

dxeI x2

21

0

2

dyeI y

2

4

2

0 041)(

41

412

222

22

I

rdrdedxdye

dyedxeI

ryx

yx

Classroom exercise:Finish the last step.

r

θx

y

rr

rJ

ryrx

yry

xrx

cossin

sincos

sin,cos

2 2

2 2 2 2 2 2

2 2 2

0 0

2 ( )

0 0 0 0 0 0

22 21

20 0 0 0 0

0

2

|

must be positive

x y

x y x y x y

r r r u

u

I e dx e dy

I e dx e dy e e dxdy e dxdy

e rdrd e dr e d r e du

e

I I I

x

y

r

r+drdθ

2 2 2

cos , sin

cos sin

sin cos

cos sin sin cos cos sin

cos sin

sin cos

x xr

y yr

x xry yr

x r y r

dx dr d dr r d

dy dr d dr r d

dxdy drdr r drd r d dr r d d

dxdy Jrdrd

rJ r

r

dxdy = |J|rdrdθ

2 2

2 2 2 2

2 2 2 2

2

0 0

2

0 0 0 0

2( ) 1

4 40 0 0 0 0

204 4 4 4

0 0

12

|

Since I must be positive, I=

x y

x y x y

x y r r

r u u

I e dx e dy

I e dx e dy e e dxdy

e dxdy e rdrd e rdr

e d r e du e

Change of Variables: General Cases

31 2

1 2 3

1 1 1 2 2 2 1 2 1 2

1 2 1 2

1 1 2 2 1 2 1 2

( , ,..., ), ( , ,..., ),..., ( , ,..., )

({ }), , 1,2,3,...,

... ( , ,..., ) ...

... ( ( , ,..., ), ( , ,..., ),..., ( , ,..., )

n

n

n n n n n

i i j

b bb b

n n

a a a a

n n n n

x x u u u x x u u u x x u u u

x x u i j n

f x x x dx dx dx

f x u u u x u u u x u u u

' '' '31 2

' ' ' '1 2 3

' '' '31 2

' ' ' '1 2 3

1 11 2

1 21 1 1

1 2

2 2 2

1 2

1 2

1 2

) | | ...

... ( ({ })) ...

......

...,or

... ...

| |

... ...

...

n

n

n

n

n

n

n

n n n

b bb b

n

a a a a

b bb b

i j n

a a a a

x x xx xu uu u u

xx xu u u

xx xu u u

J du du du

f x u J du du du

J J

1

2 2 2

1 2

1 2

...,or, { }

... ... ... ...

...

n

in

j

n n n

n

xu

x x xxu u uu

x x xu u u

J

Functions in 3 Dimensions

x

y

z

P(x,y,z)

x

y

zO

P(r,θ, )

r

x

y

z

cos

sinsin

cossin

rz

ry

rx

0tan

0tan

cos

1

1

1

2222

x

x

zyxr

xy

xy

rz

Integrals in 3 Dimensions

2

( , , ) ( , , )

( , , ) sin

V V V

V

M dm x y z dV x y z dxdydz

r r drd d

Separation of Variables

)()()(),,(

)()()(),,(

)()()(),,(

wWvVuUwvu

rRr

zZyYxXzyxf

General Curvilinear Coordinates

),,(),,,(),,,(

),,(),,,(),,,(

332211

321321321

zyxqqzyxqqzyxqq

qqqzzqqqyyqqqxx

zz

y

x

sin

cos

Cylindrical polar coordinates:

Classroom exercise: write the volume elementIn cylindrical polar coordinates.

P(ρ, ,z)

r

x

y

z

Fields: Scalar, Vector and Tensor

),,( pppp zyxfT

kzyxhjzyxgizyxfv pppppppppp ),,(),,(),,(

kzyxwjzyxvizyxuF pppppppppp ),,(),,(),,(

),,( pppp zyxd

kkpjkpikpkjpjjpijpkipjipiipp zzzyzxyzyyyxxzxyxxp

kkJjkJikJkjJjjJijJkiJjiJiiJJ zzzyzxyzyyyxxzxyxxp