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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;
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
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