13 ch0 preview problems 2013 s

7/30/2019 13 Ch0 Preview Problems 2013 s

1/37

PreviewPreview:: Preliminary and Engineering Problems

Binomial theoremBinomial theorem

EulerEulers formulas formula

DifferentiationDifferentiation

IntegrationIntegration

Complex variablesComplex variables

Tensor and vector calculusTensor and vector calculus

Hyperbolic functionsHyperbolic functions

Simplest Differential EquationsSimplest Differential Equations

Professor K.T. Chau

P-1


2/37

2 3

2 3

2 3

( 1) ( 1)( 2)lim (1 ) lim[1 ( ) ( ) ( ) ...]1! 2! 3!

(1 1/ ) (1 1/ )(1 2 / )lim[1 ...]

1! 2! 3!

1 ...1! 2! 3!

x n

n n

n

x n x n n x n n n xe n n n n

x n x n n x

x x x

= + = + + + +

= + + + +

= + + + +

1 2 2 1( 1)( )2

n n n n n nn nx h x nx h x h nxh h + = + + + + +

Binomial theorem

Proof of power series ofex

Review on fundamentals (something you MUST know)

0

1

2 2 2

3 3 2 2 3

4 4 3 2 2 3 4

( ) 1

( )

( ) 2

( ) 3 3

( ) 4 6 4

...

a b

a b a b

a b a ab b

a b a a b ab b

a b a a b a b ab b

+ =

+ = +

+ = + +

+ = + + +

+ = + + + +

Pascals triangle (1654)

Apianus (1527)

(1261)

Jia XianPascal

Yang Hui

P-2


3/37

1 2 2 1

1 1 2 2 2 1 1

( 1)( )

2

...

n n n n n n

n n n r n r r n n nn n n n

n nx h x nx h x h nxh h

x C x h C x h C x h C xh h

+ = + + + + +

= + + + + + + +

( 1)...( 1) !

! !( )!r

n

n n n r nC

r r n r

+= =

Example P-1

1

1 1

( 1)! ( 1)!

( 1)!( )! !( 1 )!

( 1)![ ( )]

( 1)!( )!

!!( )!

r r

n n

rn

n n

C C r n r r n r

nr n r

r n r

n Cr n r

+ = +

= +

= =

Basic identity

Related to probability

Problem P-1 Find S?

1 2 11 ... 1r nn n n nS C C C C = + + + + + + +

P-3

Hua Luogeng


4/37

Review on compound interest and the birth ofe

Problem P-2 FindS1?1 2 1 1

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

n n nS C C C = + + + +

Growth of money in bank

(1 )S P r= + r = annual interest ratePr= interest at the end of 1 year

After year, take out money and redeposit (a bigger P)

2[ (1 )](1 ) (1 )2 2 2r r rS P P= + + = +

After 1/n year, take out money and redeposit (a bigger P)

[ (1 )...](1 ) (1 )nr r r

S P Pn n n= + + = +

The best you can earn in compound interest (in 1 year)

Compound interest

lim (1 )n rn

rS P Pe

n= + =

1lim (1 ) 2.71828...nn

en= + =

P-4

Euler

Eulers number


5/37

Gerolamo Cardano

(1501-1576)

Italian mathematician Gerolamo Cardano is the firstknown to have introduced complex numbers in 1545.He called them "fictitious", during his attempts to find

solutions to cubic equations in the 16th century

History of Complex Numbers

3 2 0ax bx cx d + + + =

1i =

General formula of roots for cubic equation

P-5


6/37

Review on Eulers formula

(cos sin )ix iy re r i + = = +

Demoivres Formula

Example P-2 Special case ofr=1 and =

1ie = 1 0ie + = Eulers formula

Problem P-3 Find ?ii =(Answer: Infinite answers and the smallest one is 0.207879576)

Problem P-4 Find ?i i =(Answer: Infinite answers and the smallest one is 4.810477381)

1i =

(cos sin ) cos sinni n i n + = +

P-6

LHopitals Rule (actually by Johann Bernoulli)

( ), ( )f x g x

( ), ( ) 0f x g x

( ) ( )

lim lim( ) ( )x a x a

f x f x

g x g x

=

x a( ), ( )f x g x

are differentiable

0,

0

{ }( ) ( 1) ( 2) (1) ( 3) (2) ( 1) ( 1) ( )

... ( 1) ( 1)k k k k k k k k d

VU U V U V U V UV UV dx

= + + +

LHopitalBernoulli


7/37

Swiss mathematician and scientist

Euler (1707-1783)

Saturday afternoon lessons from

Johann Bernoulli

Johann Bernoulli

(1667 1748)

Basel University(1459)

the oldest universityin Switzerland

Euler cannot get a job

at Basel University

Daniel Bernoulli

Father-son

friends

Teacher & inspirer

Nicolaus II Bernoulli

Catherine I

Russia

1727

Frederick II

Prussia

1741

P-7


8/37

Swiss franc

P-8


9/37

Letters of Euler on different Subjects in Natural Philosophy

Addressed to a German Princess

http://www.math.dartmouth.edu/~euler/tour/tour_00.html

886 publications of Euler available here

German Princess

P-9

Seven Bridges of

KnigsbergGraph theory


10/37

Review on differentiation

( ) lnu ud dua a adx dx=1log lnad duudx u a dx=1lnd duudx u dx=u ud du

e edx dx=

Derivatives of Exponential and Logarithmic Functions

( ) 1n nd

nx

dx

=( ) 0d

cdx

=( )( ) ( )

0

limh

f x h f xdff x

dy h

+ = =

( )d dv du

uv u vdx dx dx

= +

DefinitionPower rule

( )d du

cu cdx dx

=

2

du dvv u

d u dx dx

dx v v

=

Quotient ruleProduct rule

( )sin cosd du

u udx dx

= ( )cos sind du

u udx dx

=

Circular functions (Trigonometric Functions )

( ) 2tan secd du

u udx dx

=

Chain Rule: dy dy dudx du dx

=

(sinh ) coshd du

u udx dx

= (cosh ) sinhd du

u udx dx

= 2(tanh ) sechd du

u udx dx

=

Hyperbolic functions

( ), ( )y y u u u x= =

1

2

1sin

1

d duu

dx dxu

=

1

2

1tan

1

d duu

dx u dx

=+

1

2

1cos

1

d duu

dx dxu

=

1

2

1(sinh )

1

d duu

dx dxu

=+

1

2

1(cosh )

1

d duu

dx dxu

=

1

2

1(tanh )

1

d duu

dx dxu

=

( ) 1n nd du

u nu

dx dx

=

( )d du dv

u vdx dx dx

+ = +

Sum rule Constant multiple

P-10


11/37

Leibniz's rule of differentiation under integral sign

( ) ( )

( ) ( )

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

h x h x

g x g x

d df x dh x dg xf x d d f x h x f x g x

dx dx dx dx

= +

General Leibniz rule

0

( )n k n k n

k

nn k n k k

d d f d g fg C

dx dx dx

=

=

Example P-3 9 9 8 2 79 9 8 2 7

sin sin 9 8 sin( sin ) 9 ( ) ( ) ...2!

cos 9sin

d d x d d x d d xx x x x xdx dx dx dx dx dx

x x x

= + + += +

0

Partial differentiation0

0

( , ) ( , )lim ( )

( , ) ( , )lim ( )

y xx

x yy

f f x x y f x y ff

x x x

f f x y y f x y f

y y y

+ = = =

+ = = =

( , ) ( , )

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

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

0 0

and

as and , the total differential i

x x x y y y f f f

f f x x y y f x y

f x x y y f x y y f x y y f x y

f x x y y f x y y f x y y f x yx y

x y

x y df

+ + +

= + +

= + + + + + + + + +

= +

s

f fdf dx dyx y

= +

Total differential

f f

df dx dyy

= +

Proof

1 2

1 2

... nn

f f fdf dx dx dx

x x x

= + + +

P-11

Leibniz


12/37

( , ) ( ) ( )and ,f f x y x x u y y u

f df f dx f dydf dx dy

x y du x du y du

= = =

= + = +

Chain rule

Problem P-5 Show that

Polar coordinates and, Cartesian coordinates x and y, x=cos ,y=sin, transform into one in and

2 2 2

1 2 1 2 1 2 2 1 1 2

( ) 2 2v u v u v u

uv u vx x x x x x x x x x

= + + +

2 2 2

2 2 1/2

21

2 2 2 2

2

2

cos sin( )

/ sin sin costan

1 ( / )sin cos

cos sin

( )

, ,

), ,

,

and

xx y

x x y y

y x y (y / x

x y x x y y

x x x y

f f

x x x

= + = = =

+

= = = = = =

+ +

= + = = +

=

2 2 2 22

2 2 2 2 2 2

1 1( ) ( , )

2 f f f f f ff x y

y y y x y

= = + = + +

2 2

2 2

f

x y

+

( , ) ( , )f x y f

Example P-4

P-12


13/37

( ) ,!

)()(

10

0)(

=

=n

nn

xxn

xfxfTaylor series expansion

"++=!6!4!2

1)cos(642 xxx

x "++=!7!5!3

)sin(753 xxx

xx"++++=!3!2

132 xx

xex

MaClaurin series ( )( )

1

(0)( )!

nn

n

ff x xn

=

=

(4) (4)

( ) sin (0) 0

'( ) cos '(0) 1

''( ) sin ''(0) 0

'''( ) cos '''(0) 1

( ) sin (0) 0

f x x f

f x x f

f x x f

f x x f

f x x f

= =

= =

= =

= = = =

2 3

3 5 7

2 1 3 5 7

0

'(0) ''(0) '''(0)(0)

1! 2! 3!

3! 5! 7!

sin ( 1)(2 1)! 3! 5! 7!

n

n

n

f f ff x x x

x x xx

x x x xx xn

+

=

+ + + +

= + +

= = + + +

Example P-5 ( ) sinf x x=

Examples

Expand

Example P-6

2 2 3 3 4 4 5 5

2 4 3 5

12! 3! 4! 5!

1 ( )2! 4! 3! 5!

ix i x i x i x i xe ix

x x x xi x

= + + + + + +

= + + + + +

"

" "

Second approach

Euler formula

"++=!6!4!2

1)cos(642 xxx

x

"++=!7!5!3

)sin(753 xxx

xx

( ) xf x e=

( ) ( ) ( ) ( ) ... xf x f x f x f x e = = = = =

2 3

2 3

'(0) ''(0) '''(0)(0)

1! 2! 3!

12! 3!

x f f fe f x x x

x xx

= + + + +

= + + + +"(0) (0) (0) (0) ... 1f f f f = = = = =

cos sinixe x i x= +

All 3 formulas are derived

P-13


14/37

Review on integration

Multiple integral

2b

aV y dx= = 2

d

cV x dy

u dv uv v du= Integration by parts

Definition Summing area under a function

Finding volume

P-14

( ) ( 1) ( 2) ( 3) ( )...( 1)n n n n n ng dx f g f g f g fg dx = +

Generalized integration by parts


15/37

General rule of integration

1( ) ( )ax dx f u du

a

=

( ){ ( )} ( )

( )

dx F uF f x dx F u du du

du f x= =

( )u f x=

1

1

n

n uu dxn

+

=+ 1n

1lndu u

u=

u ue du e=

lnln

ln ln

u a uu u a e a

a du e du a a= = = 0, 1a a>

Transformation rule

1( ) ( )F ax b dx F u du

a+ = u ax b= +

2( ) ( )F ax b dx uF u dua

+ = u ax b= +1( ) ( )nn

nF ax b dx u F u du

a

+ = nu ax b= +

2 2( ) ( cos )cosF a x dx a F a u udu = sinx a u=2 2 2( ) ( sec )secF a x dx a F a u udu+ = tanx a u=2 2( ) ( tan )sec tanF x a dx a F a u u udu =

secx a u=

P-15


16/37

1 ( )( )ax

F ue dx du

a u=

axx e=

(ln ) ( ) uF x dx F u e du= lnu x=1(sin ) ( )cosx

F dx a F u udua

= 1sin

xu

a

=

2

2 2 2

2 1(sin ,cos ) 2 ( , )

1 1 1

u u duF x x dx F

u u u

=+ + + tan 2

xu =

Definite integrals

( ) ( ) ( ),

b

a f x dx b a f c= a c b<

( )1

! ( )( )

2 ( )

n

nC

n f zf z dz

i z a +=

>Remarkable results: Value of

f(z) and its higher derivatives

only depends on boundary

values on C

Singular points

( )f a is not analytic then a = isolated singular point

Poles

( )( )( )n

zf zz a

= z = a is a pole of ordernz = a is a simple pole ifn =1

( ) 0a

Example P- 82

( )( 3) ( 1)

zf z

z z=

+has two singularities, a pole of order 2 at z=2

and a simple pole atz= 1.

(orf(z) has singularity at a)

P-18


19/37

21 10 1 21

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

( )( ) ( )

kn nkn n

k

a a af z a a z a a z a a z a

z az a z a

+

=

= + + + + + + + =

Principal part Analytic part

This the principal part has infinite terms, it is an essential singularity

Example P-91/

2

1 11 ...

2!

zez z

= + + +

For simple pole

1

1 1

1lim {( ) ( )}

( 1)!

nn

nz a

da z a f z

n dz

=

a1 = residue

1 lim( ) ( )z a

a z a f z =

has is an essential singularity at z = 0

Residue Theorem

1 1 1( ) 2 ( ...)

Cf z dz i a b c

= + + +

>

C

ab

c

Laurents series

Residue

P-19


20/37

Branch point & Branch cut nw z=

z planew plane

2

n

Branch cut

0-x is a branch cut (restrict to ensure single-valuedness)

1/ /n i nw e =

1/n

2

/ n

iz e =

n different points in w-plane corresponds

to the same point inz-plane

0 1 1{ } ,{ } ,...,{ }n n n

nz z z

nbranches of single-valued functions

Removable singularity

3 5 2 4

2 4

sin sin( ) sin 1( ...) 1 ...

3! 5! 3! 5!

( ) ( )1 ...3! 5!

z u u u u u uu

z u u u

z z

+= = = + = + +

= + + +

z u =Example P-10

(no singularity)

P-20


21/37

Example P-11 Show that1

0, 0 1

1 sin

px

dx px p

= <

Rr

y

xBE

GHJ

1

C

D

1 1 2 1 12 0( 1)

20 2

(R ) ( ) ( )2

1 1 R 1 1

p i p i i p i p iR rp i

i i ir R

x e iR d xe dx re ir ddx ie

x e xe re

+ + + =+ + + +

idz iR d

=

idz ir d

=

, 1 ... 0

BDEFG

R p < 0, 0 ... 0HJA

r p >

1 2 ( 1) 10( 1)

02

1 1

p i p pp ix e x dx

dx iex x

+ =

+ + 1

2 ( 1) ( 1)

0[1 ] 2

1

pi p p ixe dx ie

x

=+

1 ( 1)

2 ( 1)0

2 2

1 1

sin

p p i

i p p i p i

x ie idx

x e e e

p

= =

+

=

QED

P-21


22/37

First rank or order(e.g. displacement, velocity)

zeroth rank or order(e.g. temperature, pressure)

Second rank or order(e.g. stress, strain)

Third rank or order(e.g. permutation tensor )

1850 1919

Woldemar Voigt

Elementary Tensor Analysis

Fourth rank or order(e.g. elastic tensor )

Scalar (independent of direction)

3

1 1 2 2 3 3

1

u e e e e ei i i ii

u u u u u

== + + = =

1st order (vector) (e.g. displacement)

Einstein notation

(drop summation)

3 3

1 1

e e e eij i j ij i ji j

= =

= =

2nd order (e.g. stress, strain)

2nd order tensor (e.g. stress)

There are two direction senses

1x

2x

3x

3e1e

2e

ij

Plane i Directionj

Plane 1

Plane 3

3eNormal vector on plane 3 is

Plane 2

3 3 3 3

1 1 1 1

e e e e e e e eijkl i j k l ijkl i j k l

i j k l

C C

= = = =

= =

C

4th order tensor (stiffness tensor)

We need 9 components to fullydescribe stress at a point!!

1,2,3; 1,2,3i j= =

3 3 9 =

3 components to fully describe a vector

3 3 3 3 81 =

P-22


23/37

Heaviside O.

(1850-1925)Riemann, G.F.B.

Vector analysis

Gibbs, J.W.

(1839-1903)

Riemann Hypothesis

The Clay Mathematics Institute (Cambridge, Massachusetts)

$1 million award

This formula says that the zeros of the Riemann zeta function controlthe oscillations of primes around their "expected" positions.

u(u1,u2,u3)

e2

e3

e1

3

1 1 2 2 3 31

u e e e e ei i i ii

u u u u u== + + = =

(1826 1866)Age =39

odd

even

13

2

1( )( )( )2ijke i j j k k i=

This is not a tensor equation!

123 231 312 1e e e= = =

132 213 321 1e e e= = =

Permutation tensor

Even permutation

Odd permutation

133 221 131 0e e e= = =

Story

1st PhD in USA in

engineering in 1863

P-23


24/37

A Brief Review of Vector Analysis

0|cos ( ) | u v | = | u | | vdot product

i j ij =e eKronecker delta

Leopold Kronecker

(1823 1891)

German

3

1 1 2 2 3 3

1

k k k k k

u v u v u v u v u v

=

= + + = =u v

1 1 2 2 3 3 1 1 2 2 3 3= e + e + e , = e + e + eu u u v v vu v

| || | sin (0 / 2) = = w u v u v

cross product

i i ijk j k iw = e u v=w e e

Repeated indices in component

form means summation

vue=w kjijki

Polyadic form

component form

kis a dummy index (repeated)

jis a dummy index (repeated)

iis a free index (not repeated)

Rule in tensor notationRule in tensor notation

i imn m nw e u v=

same

Balance in free index on both size of =. 0,

1,

ij i j

i j

=

= =

P-24


25/37

ikjijkii vue=w ee

u v = (v u)

= + + u (v w) u v u w

u u = 0

=

=

=

1 2 3

2 3 1

3 1 2

e e e

e e e

e e e

1e

2e

3e

( )k k k = u v = u v u v

2 3 2 3 1 3 1 1 3 2 1 2 2 1 3( ) ( ) ( )e e e= u v u v u v u v u v u v = + + w u vProblem P-6 Show that these are the same

1 2 3

1 2 3

1 2 3

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

e e e

u v

e e e

u u uv v v

u v u v u v u v u v u v

=

= + +

1 2 3det( )ij i j k ijke A A A=

ijk irs jr ks js kr e e = e-

identity

determinant

P-25


26/37

Derivatives of tensors

=ix

i

e = (x,t)

ij ijii, j ij, k ij, j

j k j

v = , = , =vx x x

comma-subscript convention

Vector differential operator

,grad i iix

= = =

ie e

,div i iv= =v v

,curl ijk k jv e v= = iv e

2,ii = =

( )fg f g g f= + 2 2 2( ) 2( ) ( )fg f g f g g f = + +

( ) ( )f f f = +v v v 2( )f g f g f g = +

Identities exist for the differential operator

( ) 0f =( ) 0 =v

( ) ( = a b a) b a b ( )f f f = + v v v

2( ) ( = v v) v 2 2 )( = a a

2 2( ) ( ) = 2 2( ) 2 = a r a + r a 2 2( ) 2 = + r r

Gradient

Curl

Divergence

Laplacian

Vector calculus P-26

Useful formulas

1 2 31 2 3

=x x x

+ +

e e e

P 27


27/37

( ) ( )k i j jk

= =e e e eij ij,i x

, ,( ) ( ) ( )i j k i j k k = =e e e e e e e ejk jk i llij jk ii

ex

=

n T T

VS

dS dV =

The Divergence Theorem

( )C S

d = dS T s T nStokes Theorem.C

S

n

dSds

V

S

n

dS

Some Formulae in Cylindrical Coordinate

1 2 3 cos sin1 2 3 1 2 3r + + + +e e e e e ex x x r r z= = 1

r=e

h x

x

=h

r

er

er

ez

r

11 1

cos sin sin cos

13

re

r

z

= = + , = = +

rh h

= =zh

r 2 1 2

z

r re e e e e e

e

rr

ee= , =e e

P-27

S f l i li d i l di t P 28


28/37

( )( )

1 1(

1( )

r z r z z

r r z z z z z r

r z r r

= + + + e +e e e e er z

= + + + + +e e e e e e e e e e e e

+ +e e e e e e

r

r z z r r

z r

u u ur

u uu u u u+ )u

r r z r z z

uu uu

r r r

+

u

1( )

2 = +u u z r r

zz rr

1u u u u= , = , = +

z r r r

1 1 1 1 1( ) ( ) ( )2 2 2

r z r z r z rz

u u uu u u u= + , = + , = +

r r r r z z r

1 1r zr

uu uu

r r r z

= + + +

u

1 1( ) ( ) ( )r zu = +e e e

z r z ru u uu u u u+r z z r r r r

+

2 2 2

2

2 2 2 2

1 1f f f ff = f = + + +

r rr r z

Some formulas in cylindrical coordinate P-28

P 29


29/37

011

2=++ u

r

u

r

u rrr

1

x y r r

= + = + x y re e e e

2 1 1( ) ( )u uu ur r r r

= = + + r re e e e

2 22

2 2( ) ( ) ( )

u uu u u

x y x y x y

= = + + = +

x y x ye e e e

=

re er

=

ee

d

2

2

2 2

2 2

2 2

2 2 2

2

2 2

1( )

1 1 1 1 1 1 1( ) [ ( ) ( ) ]

1 1 1 1[ ]

1 1

u u u

r r r r

u u u u u ur r r r r r r r r r

u u u u

r r r r r r

u u

r r r

+ =

+ = + + +

= + +

= +

r r

rr r

r r

e e e

eee e e e e e

e e e e e

re

e

Laplace equation in 2-D polar form

1 =r re e 0 =re e 1 =e e

Example P-12

Will be useful later!

P-29

R i H b li f ti P 30


30/37

Review on Hyperbolic functions

2sinh

xx eex

=

1csch

sinhx

x=

cosh2

x xe ex

+=

1sech

coshx

x=

x

xx

cosh

sinhtanh =

x

xx

sinh

coshcoth =

xx sinh)sinh( =

cosh( ) coshx x =

1sinhcosh 22 = xx2 2sinh cosh 1x x=

2 2cos sin 1x x+ =

tanh x

sinh

cosh x

P-30

i h h i P 31


31/37

sinhx, coshx versus sinx, cosx P-31

Lambert (17281777)

V Riccati (17071775)

P-32


32/37

sinhx, coshx versus sinx, cosx

hyperbola circle

Hyperbolicfunctions

Circular

functionsResemblance

2 21 tanh sechx x =

yxyxyx sinhcoshcoshsinh)sinh( +=+

yxyxyx sinhsinhcoshcosh)cosh( +=+

xxdxd cosh)(sinh =

xxdx

dsinh)(cosh =

2(tanh ) sechd x xdx

=

(sech ) sech tanhd x x xdx =

2(coth ) cschd x xdx

=

(csch ) csch cothd

x xdx

=

1

2

1(sinh )1

dx

dx x

=+

1

2

1(cosh )

1

dx

dx x

=

1

2

1(tanh )

1

dx

dx x

=

1

2

1(sech )

1

dx

dx x x

=

1

2

1

(coth ) 1

d

xdx x

=

1

2

1(csch )

1

dx

dx x x

= +

P-32

2sinh

xx eex

= cosh

2

x xe ex

+=

P-33


33/37

2

sinh cosh

cosh sinh

sech tanh

udu u C

udu u C

udu u C

= +

= +

= +

2

csch cot csch

sech tanh sech

csch coth

u udu u C

u udu u C

udu u C

= +

= +

= +

1

2 2

12 2

1 2

2 21 2

sinh

cosh

1tanh

1coth

2

2

if u

if u

du uC

aa u

du u Cau a

uC a

du a a

ua uC aa a

= ++

= +

+

1

2 2

1

2 2

1sec 0

1

csc 0

du u

h C u aa au a u

du u

h C ua au u a

= +

13 ch0 preview problems 2013 s

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