1.抽象代數導論 (Introduction to Abstract Algebra)-2.張量分析 (Tensor Analysis)-3.正交函數展開 (Orthogonal Function Expansion)-4.格林函數 (Green's Function)-5.變分法 (Calculus of Variation)-6.攝動理論 (Perturbation Theory)

N968200 高等工程數學 ※ 先修課程：微積分﹑工程數學 ( 一 )-( 三 )

Reference:1. Birkhoff, G., MacLane, S., A Survey of Modern Algebra, 2nd ed, The Macmillan Co, New York, 1975.

2. 徐誠浩 , 抽象代數 - 方法導引 , 復旦大學 , 1989.

3. Arangno, D. C., Schaum’s Outline of Theory and Problems of Abstract Algebra, McGraw-Hill Inc, 1999.

4. Deskins, W. E., Abstract Algebra, The Macmillan Co, New York, 1964.

5. O’Nan, M., Enderton, H., Linear Algebra, 3rd ed, Harcourt Brace Jovanovich Inc, 1990.

6. Hoffman, K., Kunze, R., Linear Algebra, 2nd ed, The Southeast Book Co, New Jersey, 1971.

7. McCoy, N. H., Fundamentals of Abstract Algebra, expanded version, Allyn & Bacon Inc, Boston, 1972.

8. Hildebrand, F. B., Methods of Applied Mathematics, 2nd ed, Prentice-Hall Inc, New Jersey, 1972..

9. Burton, D. M., An Introduction to Abstract Mathematical Systems, Addison-Wesley, Massachusetts, 1965.

10. Grossman, S. I., Derrick, W. R., Advanced Engineering Mathematics, Happer & Row, 1988.

11. Hilbert, D., Courant, R., Methods of Mathematical Physics, vol(1), 狀元出版社 , 台北市 , 民國六十二年 .

12. Jeffrey, A., Advanced Engineering Mathematics, Harcourt, 2002.

13. Arfken, G. B., Weber, H. J., Mathematical Methods for Physicists, 5th ed, Harcourt, 2001.

14. Morse, F. B., Morse, F. H., Feshbach, H., Methods of Theoretical Physics, McGraw-Hill College, 1953

David HilbertDavid Hilbert BornBorn January 23, 1862 Wehlau, East Prussia January 23, 1862 Wehlau, East PrussiaDiedDied February 14, 1943 Göttingen, Germany February 14, 1943 Göttingen, GermanyResidenceResidence GermanyGermanyNationalityNationality GermanGermanFieldField Mathematician MathematicianErdősErdős Number Number 44InstitutionInstitution University of Königsberg and Göttingen UniversitUniversity of Königsberg and Göttingen Universityy

Alma Mater University of KönigsbergAlma Mater University of KönigsbergDoctoral AdvisorDoctoral Advisor Ferdinand von Lindemann Ferdinand von LindemannDoctoral StudentsDoctoral Students Otto Blumenthal Otto Blumenthal Richard CourantRichard Courant Max DehnMax Dehn Erich HeckeErich Hecke Hellmuth KneserHellmuth Kneser

Robert KönigRobert König Erhard SchmidtErhard Schmidt Hugo SteinhausHugo Steinhaus Emanuel LaskerEmanuel Lasker Hermann WeylHermann Weyl Ernst ZermeloErnst Zermelo

Known forKnown for Hilbert's basis theorem Hilbert's basis theoremHilbert's axiomsHilbert's axiomsHilbert's problemsHilbert's problemsHilbert's programHilbert's programEinstein-Hilbert actionEinstein-Hilbert actionHilbert spaceHilbert space

SocietiesSocieties Foreign member of the Royal SocietyForeign member of the Royal SocietySpouseSpouse Käthe Jerosch (1864-1945, m. 1892)Käthe Jerosch (1864-1945, m. 1892)ChildrenChildren Franz Hilbert (1893-1969)Franz Hilbert (1893-1969)HandednessHandedness Right handed Right handed

The finiteness theoremThe finiteness theorem

Axiomatization of geometryAxiomatization of geometry

The 23 ProblemsThe 23 Problems

FormalismFormalism

~ from Wikipedia~ from Wikipedia

Philip M. MorsePhilip M. Morse

““OperationsOperations research is an research is an applied science utilizing all known applied science utilizing all known

scientific techniques as tools in scientific techniques as tools in solving a specific problem.”solving a specific problem.”

Founding ORSA President (1952)Founding ORSA President (1952)

B.S. Physics, 1926, Case Institute; B.S. Physics, 1926, Case Institute;

Ph.D. Physics, 1929, Princeton Ph.D. Physics, 1929, Princeton University.University.

Faculty member at MIT, 1931-1969.Faculty member at MIT, 1931-1969.

Methods of Operations ResearchMethods of Operations ResearchQueues, Inventories, and MaintenanceQueues, Inventories, and MaintenanceLibrary EffectivenessLibrary EffectivenessQuantum MechanicsQuantum MechanicsMethods of Theoretical PhysicsMethods of Theoretical PhysicsVibration and SoundVibration and SoundTheoretical AcousticsTheoretical AcousticsThermal PhysicsThermal PhysicsHandbook of Mathematical Functions, with Formulas, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical TablesGraphs, and Mathematical Tables

Francis B. HildebrandFrancis B. Hildebrand

George ArfkenGeorge Arfken

Introduction to Abstract AlgebraIntroduction to Abstract Algebra抽象代數導論抽象代數導論

• Preliminary notionsPreliminary notions• Systems with a single operationSystems with a single operation• Mathematical systems with two operationsMathematical systems with two operations• Matrix theory: an algebraic viewMatrix theory: an algebraic view

群胚封閉性

半群 單子 群

抽象代數系統 RVRRR ,,,,,,,

反元素eaaaa 11

,R

性之分佈對、

acabacb

cabacba

單子環

可交換群

可交換單子

可交換半群

可交換環 環

結合性

cba

cba

可交換性abba

向量

,V

RV

=

向量空間

,R

,,R

純量可交換單子環

單位元素aaeea

),,,(),,( RV

域

群胚 Groupoid

• A goupoid must satisfy is closed under the rule of combination

R

,R

RbaRb,a

• Ex. Consider the operation defined on the set S= {1,2,3} by the operation table below.

From the table, we see

2 (1 3)=2 3=2 but (2 1) 3=3 3=1

The associative law fails to hold in this groupoid(S, )

2 1 3* 1 2 3

1 2 3 1 3 2 3 2 1

• A semigroup is a groupoid whose operation satisfies the associative law.

(groupoid)

半群 Semigroup

cbacbaRc,b,a

RbaRb,a

• Ex. If the operation is defined on by a b = max{ a, b },that is a b is the larger of the elements a and b, or either one if a=b.

a (b c) = max{ a, b, c } = (a b) c

that shows to be a semigroup

• If and is a semigroup, then

proof.

),(R#

#R

Rdc,b,a, )(R,d)c)((bad)(cb)(a

d)(cb)(a

xb)(a

xby d)(c denoted x)(ba

d))(c(bad)c)((ba

• A semigroup having an identity element for the operation is called a monoid.

(groupoid) (semigroup)

單子 Monoid

,R

aaee aRa Re

e

RbaRb,a

cbacbaRc,b,a

Ex. Both the semigroups and are instances of monoids

for each

The empty set is the identity element for the union operation.

for each

The universal set is the identity element for the intersection operation.

),(SU ),(SU

A A A U A

A AU U A U A

群 Group

• A monoid which each element of has an inverse is called a group

(groupoid)

(semigroup)

(monoid)

,R R

RbaRb,a

cbacbaRc,b,a

aaee aRa Re

eaaaaRaRa 1-1-1

• If is a group and ,then

Proof. all we need to show is that

from the uniqueness of the inverse of

we would conclude

a similar argument establishes that

,R Rba, -1-1-1 abb)(a

eb)(a)a(b)a(bb)(a -1-1-1-1 ba

-1-1-1 abb)(a

e

aa

)a(ea

)a)b((ba)a(bb)(a

1-

1-

-1-1-1-1

eb)(a)a(b -1-1

Commutative 可交換性

group1-1-1

monoid

semigroup

groupoid

eaaaaRaRa

aaeeaReRa

cb)(ac)(baRcb,a,

RbaRba,

abbaRba,

Commutative groupoid Commutative s

emigroup

Commutative monoid

Commutative group

• Ex. consider the set of number and the operation of ordinary multiplication, and Z represents integer.

1. Closure:

2. Associate property

3. Identity element

4. Commutative property

is a commutative monoid.

Z}ba,|2b{aS

S2bc)(ad2bd)(ac)2d(c)2b(aZdc,b,a,

)2f(e)2d(c)2b(a)2f(e)2d(c)2b(a

Zfe,d,c,b,a,

2011

)2b(a)2d(c)2d(c)2b(a

Zdc,b,a,

)(S,

Ring 環• A ring is a nonempty set with two binary op

erations and on such that

1. is a commutative group

2. is a semigroup

3. The two operations are related by the distributive laws

),(R,

semigroup

groupoid

cb)(ac)(baRcb,a,

RbaRba,

a)(ca)(bac)(b

c)(ab)(ac)(baRcb,a,

R

R

)(R,

)(R,group

1-1-1

monoid

semigroup

groupoid

eaaaaRaRa

aaeeaReRa

cb)(ac)(baRcb,a,

RbaRba,

abbaRba,

• A ring consists of a nonempty set and two operations, called addition and multiplication and denoted by and , respectively, satisfying the requirements:

1. R is closed under addition

2. Commutative

3. Associative

4. Identity element 0

5. Inverse

6. R is closed under multiplication

7. Associate

8. Distributive law

),(R,

a)(ca)(bac)(b c)(ab)(ac)(ba 8.

cb)(ac)(ba 7.

Rba 6.

0(-a)a Ra 5.

aa00aR0 4.

cb)(ac)(ba 3.

abba 2.

Rba 1.Rcb,a,

group1-

R

Monoid Ring 單子環• A monoid ring is a ring with identity that is a semi

group with identity

monoid

semigroup

groupoid

group1-1-1

monoid

semigroup

groupoid

aaeeaRe

a)(ca)(bac)(b

c)(ab)(ac)(ba

cb)(ac)(ba

Rba

eaaaaRa

aaeeaRe

cb)(ac)(ba

Rba

abbaRcb,a,

Ring

Monoid ring

),(R,

• Ring with commutative property

abb a

a)(ca)(bac)(bc) (ab)(ac)(b a

cb)(ac)(b a

Rb a

eaaaaRa

aaeeaRe

cb)(ac)(b a

abb a

RbaRa,b,c

---

111

),(R,

Commutative

Commutative monoid Ring

aaeeaRe

a)(ca)(bac) (b

c)(ab)(ac)(ba

cb)(ac)(ba

Rba

eaaaaRa

aaeeaRe

cb)(ac)(ba

Rba

abbaRb,ca

1-1-1

,

Subring 子環• The triple is a subring of the ring

1. is a nonempty subset of

2. is a subgroup of

3. is closed under multiplication

),(S, ),(R, S

)(S,S

)(R,R

Sba

eaaaaSa

aaeeaSe

cb)(ac)(ba

Sba

abbaSa,b,c

RS

1-1-1-

• The minimal set of conditions for determining subrings Let be ring and Then the triple is a subring of if and only if

1. Closed under differences

2. Closed under multiplication

• Ex. Let then is a subring of , since

This shows that is closed under both differences and products.

RS ),(R, ),(S, ),(R,

Sba

Sb-aSba,

Z}ba,|3b{aS ),(S, numbers real of a set isR),,,(R # #

integers of setthe is ZZ,dc,b,a,

S

S3ad)(bc3bd)(ac)3d(c)3b(a

S3d)-(bc)-(a)3d(c-)3b(a

Field 域• A field is a commutative monoid ring in whic

h each nonzero element has an inverse under ),(F,

Definition of FieldDefinition of Field

cabacbaF

F

F

FF

F

)(

},0{

,

,,

,c b,a, elements of triple each For (3)

1;identity withgroup, ecommutativ a is )( (2)

0;identity withgroup, ecommutativ a is )( (1)

that such tion,multiplica and addition called , on and set

nonempty of consisting )( system almathematic a is fieldA

Vector 向量• An n-component, or n-dimensional, vector is an n tu

ple of real numbers written either in a row or in a column.

• Row vector

• Column vector

called the components of the vector

n is the dimension of the vector

n21 xxx ,,,

n

2

1

x

x

x

#k Rx

Vector space 向量空間

• A vector space( or linear space) over the field F consists of the following:

1. A commutative group whose elements are called vectors.

V(F))),,),(F,((V, or

group1-1-1

monoid

semigroup

groupoid

eaaaaVaVa

aaeeaVeVa

cb)(ac)(baVa,b,c

VbaVa,b

abbaVa,b

)(V,

2. A field whose elements are called scalars.

eaaaaFa

aaeeaFe

a)(ca)(bac) (b

c)(ab)(ac)(ba

cb)(ac)(ba

Fba

eaaaaFa

aaeeaFe

cb)(ac)(ba

Fba

abbaFb,ca

1-1-1

1-1-1

,

),(F,

3. An operation 。 of scalar multiplication connecting the group and field which satisfies the properties

1 2 1 2

1 2 1 2

(a) and , there is defined an element ;

(b) ( ) ( ) ( );

(c) ( ) ( ),

(d) ( ) ( ) ( );

(e) 1 , where 1 is the field identity element.

c F x V c x V

c c x c x c x

c c x c c x

c x y c x c y

x x

V is closed under left multiplication by scalars

nmijij

nmij

nmnm

Mcaac

MaRc

nm

MM

by tionmultiplica scalar define , and For

addition.matrix of operation the is and matrices all of

set the is where, be group ecommutativ the Let

)()(

)(

),(

#

← Vector Space

When m = n, we denote the particular vector space by Mn(R#)

• Ex:

tion.multiplica scalar under closed is (2)

of subgroup a is (1)

W

VW );,(),(

Subspace 向量子空間

WVW ,

• Let V(F) be a vector space over the field F

W(F) is a subspace of V(F)

The minimum conditions that W(F) must satisfy to be a subspace are:

.

;,

WcxFcWx

WyxWyx

imply and

implies

• If V(F) and V’(F) are vector spaces over the same field, then the mapping f : V → V’ is said to be operation-preserving if

),()(

),()()(

xcfcxf

yfxfyxf

. and , elements of pair FcVyx

f preserves

V(F) and V’(F) are algebraically equivalent whenever there exists a one-to-one operation-preserving function from V onto V’

Linear Transformations 線性轉換

• Let V and W be vector spaces. A linear transformation from V into W is a function T from the set V into W with the following two properties:

.),()(

.,),()()(

scalars and (ii)

(i)

VxxTxT

VyxyTxTyxT

•x •T(x)T

V W

T is function from V to W, }|)({ VxxT

Let V and W be vector spaces over the field F and let T be a linear transformation from V into W.

• The null space (kernal) of T is the set of all vectors x in V such that T(x) = 0

ker { | ( ) 0}T x V T x

• If V is finite-dimensional, the rank of T is the dimension of the range of T and the nullity of T is the dimension of the null space of T.

•

••

•

• 0

T

ker T

V W

ran T

UxxTSxTS in for )),(())((

The Algebra of Linear Transformations

•Let T : U → V and S : V → W be linear transformations, with U, V, and W vector spaces.

The composition of S and V

1 2

1 2 1 2

1 2

and are vectors in , then

( )( ) ( ( )) (by definition of )

( ( ) ( )) (by linearity of )

if x x U

S T x x S T x x S T

S T x T x T

1 2

1 2

( ( )) ( ( )) (by linearity of )

( )( ) ( )( ) (by definition of )

Similarly, we have, with in and a scalar,

( )

S T x S T x S

S T x S T x S T

x U

S T

( ) ( ( )) (by definition of )

( ( )) (by linearity of )

( ( )) (by linearity of

x S T x S T

S T x T

S T x

)

( )( ) (by definition of )

S

S T x S T

Representation of Linear Transformations by Matrices線性轉換的矩陣表示

Let V be an n-dimensional vector space over the field F. T is a linear transformation, and α1, α2,…,αn are ordered bases for V. If

A

TTTT

aaaT

aaaT

aaaT

n

nn

nnnnnn

nn

nn

),,,(

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

)(

)(

)(

21

2121

2211

22221122

12211111

nnnn

n

n

aaa

aaa

aaa

A

21

22221

11211

其中稱 A 為 Linear Transformation T在 α1, α2,…,αn 下的矩陣

Inner Product Inner Product 向量內積向量內積

),(),( (4)

),(),(),(

),(),(),( (3)

),(),( (2)

0 ifonly and if 0),(

0),( (1)

scalars real are and and in vectors are , ,

.definition

the fromy immediatelfollow product inner the of properties Certain

by denoted is which),( written, and of product inner the

in vectors two be and Let

3

T

abba

cbcacba

cabacba

baba

aaa

aa

Rcba

ba

bababa

Rba 3

,

][ 332211

3

2

1

321

321321

bababa

b

b

b

aaa

kbjbibkajaia

It follows from the Pythagorean theorem that the length of the vector It follows from the Pythagorean theorem that the length of the vector

cos2

cos),(

*

),(),(

222

2/123

22

21

23

22

21321

babaab

baba

Rba

aaaaa

aa

a

them between angle the be let and in vectors nonzero be and Let

by denoted is vector the of length The

is

3

aaa

aaakajaia

aa

xx

zz

yy

23

22

21 aaa

22

21 aa

a11

a2

a3

xx

yy

baθθ

|b - a|

Inner Product Space 向量內積空間

yxx,yyx

V

xyyx

yxyx

yxyx

zyzxzyx

zxyxzyx

xx,x

x,x

Vx,yVyx

V

nn

nn

T)( and

an constitute to said is product, inner its withtogether , space vector The

)( (4)

(3)

(2)

ifonly and if )(

)( (1)

.properties following the

has it if , on product inner an be to said is )( number real a in and

vectors of pairevery to assigns that function aspace, vector real a is If

2211

11

),(,

),(),(

),(),(

),(),(),(

),(),(),(

00

0

space product inner

Eigenvalues and Eigenvectors特徵值與特徵向量

0. plus for T of orseigenvedct the all of consisting

set the is eigenvalue the for of the then , of eigenvalue an is If (c)

. whichfor vector nonzero

any is eigenvalue the for T of an hen T, of eigenvalue an is If (b)

in vector nonzero some for whichfor scalar a is of An(a)

. space vector a on operator linear a be Let

,

λv}V|T(v){v

T eigenspaceT

λvT(v)v

reigenvecto

VvλvT(v)Teigenvalue

VVT:V

rs.eigenvecto the

of multiples scalarby determined origin the through lines the fixes

operator linear the case, this In ly.respective , and rseigenvecto

ingcorrespond withT of seigenvalue are 2 and 3 that follows it

and Since

by defined on operator linear the be Let Ex. 2

T

TT

xxTT

2

1

1

1

,2

12

2

1

12

14

2

1

1

13

1

1

12

14

1

1

.12

14)(R

11 11 22 33

11

22 I + 2 jI + 2 j

I + jI + j 11

22

33

44 2 I + 4 j2 I + 4 j

3 I + 3 j3 I + 3 j

TT

Diagonalization 對角化

A square matrix is said to be a diagonal matrix if all of its entries are zero except those on the main diagonal:

n

00

00

00

2

1

A linear operator T on a finite-dimensional vector space V is diagonalizable if there is a basis vector for V each vector of which is an eigenvector of T.

1

1

232131332211332

22

12122

21112121

12

1222

1

1111

s21

)eu,(eu v )u(

u e

form in theset required

theofmember th thedeterminesfinally process thisofon continuatiA

)eu,(e)eu,(eu v eeu v, )u(

u eLet

)eu,(eu v

)u,(e or 0)e,(e-)u,(e)v,(e

iondeterminat the toleads e toorthogonal be t that vrequiremen The

eu vand u vector second a choose Then we

)u(

u e length itsby it divide and , u Let v

vectors,original theof oneany select first We vectors.original theof

nscombinatiolinear s ofset orthogonalan u,,u,u t vectorsindependen

linearly ofset a from form tosection, preceeding in the as ,desireable isIt

s

kkskss

s

ss where

l

s

ccl

cc

c

l

s

Orthogonalization of Vector Sets 向量的正交化

Gram-Schmit orthogonalization procedureGram-Schmit orthogonalization procedure

A xx yx,

matrixsymmetric a is A y, A x :form the in writtenbe can equations of set The

:equations the obtain we, write weIf

a calles is

form the of degree second of expression shomogeneouA

T

ij

nnnnnn

nn

nn

ii

nnnnnnn

A

a

yxaxaxa

yxaxaxa

yxaxaxa

x

Ay

formquadraticxxaxxaxxaxaxaxaA

)(

][

2

1

.222

2211

22222121

11212111

1,13113211222

2222

111

Quadratic Forms 二次形式

Equivalent

Equivalent

jiij

ijij

aa

yxa

A A x = yx = y

Canonical Form 標準形式

QA Q A'equation theby defined is matrix A'new the where

x'A'x' or x'QA Q x' x'QA ) x'(Q

x'Q x equation theby x'of terms in expressed be x vector the Let

T

TTTT

AA

↑↑Diagonal matrixDiagonal matrix

If the eigenvalues and corresponding eigenvectors of the real symmetric matrix If the eigenvalues and corresponding eigenvectors of the real symmetric matrix A are known, a matrix Q having this property can be easily constructedA are known, a matrix Q having this property can be easily constructed

nnnn ee AeeA ,,1111

nnnn

n

n

eee

eee

eee

21

22212

12111

Q

Let a matrix Q be constructed in such a way that the elements of the unit vectors Let a matrix Q be constructed in such a way that the elements of the unit vectors ee11, e, e22,….,e,….,enn are the elements of the successive columns of Q: are the elements of the successive columns of Q:

eigenvalueeigenvalue

eigenvectoreigenvector

nn

n

n

n

n e

e

e

e

e

e

e

e

2

1

1

12

11

1

Matrix Orthogonal I QQ or QQ

QA Q result the obtain weQby above equation the

of members equal the yingpremultiplby and exists, Q inverse the Thus

0 |Q|that follows it t,independenlinearly are e,.....,e vectors the Since

Q Q A or QA

T1-T

1-1-

1-

n1

][

00

00

00

2

1

2211

2222121

1212111

iji

nnnnnn

nn

nn

eee

eee

eee

iiiii eex A A 2'

Ex: Let T be the linear operator on R3 which is represented in the standard ordered basis by the matrix A

DAQQQ

A

200

020

001

103

011

223

1

0

2

0

1

2

3

1

3

2,2,1

463

241

665

1 have weThen

, , eigenvecor

eigenvalue

321

321

↑Diagonal matrix ↑Orthogonal matrix