N968200 高等工程數學

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

2. Reference: • Birkhoff, G., MacLane, S., A Survey of Modern Algebra, 2nd ed, The Macmillan Co, New York, 1975. • 徐誠浩, 抽象代數-方法導引, 復旦大學, 1989. • Arangno, D. C., Schaum’s Outline of Theory and Problems of Abstract Algebra, McGraw-Hill Inc, 1999. • Deskins, W. E., Abstract Algebra, The Macmillan Co, New York, 1964. • O’Nan, M., Enderton, H., Linear Algebra, 3rd ed, Harcourt Brace Jovanovich Inc, 1990. • Hoffman, K., Kunze, R., Linear Algebra, 2nd ed, The Southeast Book Co, New Jersey, 1971. • McCoy, N. H., Fundamentals of Abstract Algebra, expanded version, Allyn & Bacon Inc, Boston, 1972. • Hildebrand, F. B., Methods of Applied Mathematics, 2nd ed, Prentice-Hall Inc, New Jersey, 1972.. • Burton, D. M., An Introduction to Abstract Mathematical Systems, Addison-Wesley, Massachusetts, 1965. • Grossman, S. I., Derrick, W. R., Advanced Engineering Mathematics, Happer & Row, 1988. • Hilbert, D., Courant, R., Methods of Mathematical Physics, vol(1), 狀元出版社, 台北市, 民國六十二年. • Jeffrey, A.,Advanced Engineering Mathematics, Harcourt, 2002. • Arfken, G. B., Weber, H. J., Mathematical Methods for Physicists, 5th ed, Harcourt, 2001. • Morse, F. B., Morse, F. H., Feshbach, H., Methods of Theoretical Physics, McGraw-Hill College, 1953

3. ~ from Wikipedia David Hilbert Born January 23, 1862 Wehlau, East Prussia Died February 14, 1943 Göttingen, Germany Residence Germany Nationality German Field Mathematician Erdős Number 4 InstitutionUniversity of Königsberg and Göttingen University Alma Mater University of Königsberg Doctoral Advisor Ferdinand von Lindemann Doctoral Students Otto Blumenthal Richard Courant Max Dehn Erich Hecke Hellmuth Kneser Robert König Erhard Schmidt Hugo Steinhaus Emanuel Lasker Hermann Weyl Ernst Zermelo Known for Hilbert's basis theorem Hilbert's axioms Hilbert's problems Hilbert's program Einstein-Hilbert action Hilbert space Societies Foreign member of the Royal Society Spouse Käthe Jerosch (1864-1945, m. 1892) Children Franz Hilbert (1893-1969) Handedness Right handed The finiteness theorem Axiomatization of geometry The 23 Problems Formalism

4. Philip M. Morse Founding ORSA President (1952) B.S. Physics, 1926, Case Institute; Ph.D. Physics, 1929, Princeton University. Faculty member at MIT, 1931-1969. Methods of Operations Research Queues, Inventories, and Maintenance Library Effectiveness Quantum Mechanics Methods of Theoretical Physics Vibration and Sound Theoretical Acoustics Thermal Physics Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables “Operations research is an applied science utilizing all known scientific techniques as tools in solving a specific problem.”

5. Francis B. Hildebrand George Arfken

6. Introduction to Abstract Algebra 抽象代數導論 • Preliminary notions • Systems with a single operation • Mathematical systems with two operations • Matrix theory: an algebraic view

7. 可交換性 抽象代數系統 反元素 可交換群 單位元素 可交換單子 結合性 可交換半群 向量 封閉性 半群 單子 群 群胚 環 純量 = 可交換環 單子環 可交換單子環 域 向量空間

8. 群胚Groupoid • A goupoid must satisfy is closed under the rule of combination

9. 1 2 3 * 1 2 3 1 3 2 2 1 3 3 2 1 • 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, )

10. 半群Semigroup • A semigroup is a groupoid whose operation satisfies the associative law. (groupoid)

11. 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.

12. 單子 Monoid • A semigroup having an identity element for the operation is called a monoid. (groupoid) (semigroup)

13. 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.

14. 群 Group • A monoid which each element of has an inverse is called a group (groupoid) (semigroup) (monoid)

15. 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

16. Commutative 可交換性 Commutative groupoid Commutative semigroup Commutative monoid Commutative group

17. Ex. consider the set of number and the operation of ordinary multiplication, and Z represents integer. • Closure: • Associate property • Identity element • Commutative property is a commutative monoid.

18. Ring 環 • A ring is a nonempty set with two binary operations and on such that • is a commutative group • is a semigroup • The two operations are related by the distributive laws

19. A ring consists of a nonempty set and two operations, called addition and multiplication and denoted by and , respectively, satisfying the requirements: • R is closed under addition • Commutative • Associative • Identity element 0 • Inverse • R is closed under multiplication • Associate • Distributive law

20. Monoid Ring單子環 • A monoid ring is a ring with identity that is a semigroup with identity Ring Monoid ring

21. Ring with commutative property Commutative Commutative monoid Ring

22. Subring子環 • The triple is a subring of the ring • is a nonempty subset of • is a subgroup of • is closed under multiplication

23. The minimal set of conditions for determining subrings Let be ring and Then the triple is a subring of if and only if • Closed under differences • Closed under multiplication • Ex. Let then is a subring of , since This shows that is closed under both differences and products.

24. Field域 • A field is a commutative monoid ring in which each nonzero element has an inverse under Definition of Field

25. Vector向量 • An n-component, or n-dimensional, vector is an n tuple 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

26. Vector space向量空間 • A vector space( or linear space) over the field F consists of the following: • A commutative group whose elements are called vectors.

27. A field whose elements are called scalars.

28. An operation 。of scalar multiplication connecting the group and field which satisfies the properties V is closed under left multiplication by scalars

29. When m = n, we denote the particular vector space by Mn(R#) • Ex: ← Vector Space

30. Subspace 向量子空間 • 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:

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

32. Linear Transformations 線性轉換 • Let V and W be vector spaces. A linear transformationfrom V into • W is a function T from the set V into W with the following two • properties: • x • T(x) T W V T is function from V to W,

33. Let V and W be vector spaces over the field F and let T be a linear transformation from V into W. • 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. • The null space (kernal) of T is the set of all vectors x in V such that T(x) = 0 V W • • T • ran T • ker T • 0

34. 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

35. 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 稱A為Linear TransformationT在α1, α2,…,αn下的矩陣 其中

36. Inner Product 向量內積

37. It follows from the Pythagorean theorem that the length of the vector y z |b - a| a a2 b a3 θ a y a1 x x

38. Inner Product Space 向量內積空間

39. Eigenvalues and Eigenvectors 特徵值與特徵向量

40. 2 I + 4 j 4 3 3 I + 3 j 2 I + 2 j 2 1 I + j 1 T 1 1 2 3

41. 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: 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.

42. Orthogonalization of Vector Sets 向量的正交化 Gram-Schmit orthogonalization procedure

43. Quadratic Forms 二次形式 A x = y Equivalent

44. Canonical Form 標準形式 ↑Diagonal matrix If the eigenvalues and corresponding eigenvectors of the real symmetric matrix A are known, a matrix Q having this property can be easily constructed eigenvector eigenvalue Let a matrix Q be constructed in such a way that the elements of the unit vectors e1, e2,….,en are the elements of the successive columns of Q:

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