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9783540679950

Mathematical Foundations for Computational Engineering

by ;
  • ISBN13:

    9783540679950

  • ISBN10:

    3540679952

  • Format: Hardcover
  • Copyright: 2001-08-01
  • Publisher: Springer Verlag
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Supplemental Materials

What is included with this book?

Summary

Computational engineering is the treatment of engineering tasks with computers. It is based on computational mathematics, which is presented here in a comprehensive handbook. Engineers and scientists who deal with engineering tasks have to handle large amounts of information, which must be created and structured in a systematic manner. This demands a high level of abstraction and therefore knowledge of the mathematical foundations. From the existing rich repertoire of mathematical theories and methods, the fundamentals of engineering computation are selected and presented in a coherent fashion. They are brought into a suitable order for specific engineering purposes, and their significance for typical applications is shown. The relevant definitions, notations and theories are presented in a durable form which is independent of the fast development of information and communication technology.

Author Biography

Peter Jan Pahl is President of the International Society on Computing in Civil and Structural Engineering.

Table of Contents

Logic
Representation of Thought
1(1)
Elementary Concepts
2(3)
Propositional Logic
5(15)
Logical Variables and Connectives
5(6)
Logical Expressions
11(3)
Logical Normal Form
14(4)
Logical Rules of Inference
18(2)
Predicate Logic
20(6)
Proofs and Axioms
26(5)
Set Theory
Sets
31(3)
Algebra of Sets
34(3)
Relations
37(3)
Types of Relations
40(4)
Mappings
44(2)
Types of Mappings
46(5)
Cardinality and Countability
51(5)
Structures
56(3)
Algebraic Structures
Introduction
59(1)
Inner Operations
60(3)
Sets with One Operation
63(5)
Sets with Two Operations
68(18)
Introduction
68(1)
Additive and Multiplicative Domains
69(7)
Dual Domains
76(10)
Vector Spaces
86(13)
General Vector Spaces
86(8)
Real Vector Spaces
94(5)
Linear Mappings
99(10)
Vector and Matrix Algebra
109(22)
Definitions
109(3)
Elementary Vector Operations
112(4)
Elementary Matrix Operations
116(7)
Derived Scalars
123(4)
Complex Vectors and Matrices
127(4)
Ordinal Structures
Introduction
131(1)
Ordered Sets
132(6)
Extreme Elements
138(3)
Ordered Sets with Extreamality Properties
141(4)
Mappings of Ordered Sets
145(6)
Properties of Ordered Sets
151(7)
Ordered Cardinal Numbers
158(3)
Topological Structures
Introduction
161(2)
Topological Spaces
163(4)
Bases and Generating Sets
167(5)
Metric Spaces
172(7)
Point Sets in Topological Spaces
179(5)
Topological Mappings
184(5)
Construction of Topologies
189(12)
Final and Initial Topologies
189(6)
Subspaces
195(4)
Product Spaces
199(2)
Connectedness of Sets
201(18)
Disconnections and Connectedness
201(6)
Connectedness of Constructed Sets
207(5)
Components and Paths
212(7)
Separation Properties
219(8)
Convergence
227(32)
Sequences
227(10)
Subsequences
237(4)
Series
241(8)
Nets
249(6)
Filters
255(4)
Compactness
259(17)
Compact Spaces
259(9)
Compact Metric Spaces
268(4)
Locally Compact Spaces
272(4)
Continuity of Real Functions
276(9)
Number System
Introduction
285(1)
Natural Numbers
286(3)
Integers
289(4)
Rational Numbers
293(3)
Real Numbers
296(7)
Complex Numbers
303(4)
Quaternions
307(2)
Groups
Introduction
309(5)
Group Theory
309(3)
Outline
312(2)
Groups and Subgroups
314(5)
Types of Groups
319(23)
Permutation Groups
320(3)
Symmetry Groups
323(4)
Generated Groups
327(3)
Cyclic Groups
330(3)
Groups of Integers
333(5)
Cyclic Subgroups
338(4)
Class Structure
342(15)
Classes
342(2)
Cosets and Normal Subgroups
344(6)
Groups of Reside Classes
350(2)
Conjugate Elements and Sets
352(5)
Group Structure
357(25)
Introduction
357(2)
Homomorphism
359(7)
Isomorphism
366(7)
Isomorphic Types of Groups
373(3)
Automorphisms
376(6)
Abelian Groups
382(35)
Introduction
382(1)
Classification of Abelian Groups
383(5)
Linear Combinations
388(6)
Direct Sums
394(8)
Constructions of Abelian Groups
402(9)
Decompositions of Abelian Groups
411(6)
Permultations
417(38)
Introduction
417(1)
Symmetric Groups
418(4)
Cycles
422(6)
Conjugate Permutations
428(3)
Transpositions
431(3)
Subgroups of a Symmetric Group
434(5)
Group Structure of the Symmetric Group S4
439(11)
Class Structure of the Symmetric Group S4
450(5)
General Groups
455(27)
Introduction
455(1)
Classes in General Groups
456(8)
Groups of Prime-power Order
464(9)
Normal Series
473(9)
Unique Decomposition of Abelian Groups
482(7)
Graphs
Introduction
489(2)
Algebra of Relations
491(26)
Introduction
491(1)
Unary Relations
492(4)
Homogeneous Binary Relations
496(8)
Heterogeneous Binary Relations
504(5)
Unary and Binary Relations
509(3)
Closures
512(5)
Classification of Graphs
517(21)
Introduction
517(1)
Directed Graphs
518(6)
Bipartite Graphs
524(6)
Multigraphs
530(5)
Hypergraphs
535(3)
Structure of Graphs
538(46)
Introduction
538(1)
Paths and Cycles in Directed Graphs
539(7)
Connectedness of Directed Graphs
546(6)
Cuts in Directed Graphs
552(11)
Paths and Cycles in Simple Graphs
563(4)
Connectedness of Simple Graphs
567(2)
Cuts in Simple Graphs
569(5)
Acyclic Graphs
574(6)
Rooted Graphs and Rooted Trees
580(4)
Paths in Networks
584(61)
Introduction
584(2)
Path Algebra
586(14)
Boolean Path Algebra
600(2)
Real Path Algebra
602(1)
Minimal Path Length
602(2)
Maximal Path Length
604(2)
Maximal Path Reliability
606(1)
Maximal Path Capacity
607(2)
Literal Path Algebra
609(1)
Path Edges
609(2)
Common Path Edges
611(2)
Simple Paths
613(2)
Extreme Simple Paths
615(1)
Literal Vertex Labels
616(3)
Literal Edge Labels for Simple Graphs
619(1)
Applications in Structural Analysis
620(1)
Properties of Path Algebras
621(5)
Systems of Equations
626(1)
Solutions of Systems of Equations
626(4)
Direct Methods of Solution
630(10)
Iterative Methods of Solution
640(5)
Network Flows
645(26)
Introduction
645(2)
Networks and Flows
647(2)
Unrestricted Flow
649(4)
Restricted Flow
653(4)
Maximal Flow
657(5)
Maximal Flow and Minimal Cost
662(3)
Circulation
665(6)
Tensors
Introduction
671(1)
Vector Algebra
672(30)
Vector Spaces
672(3)
Bases
675(4)
Coordinates
679(3)
Metrics
682(4)
Construction of Bases
686(6)
Transformation of Bases
692(8)
Orientation and Volume
700(2)
Tensor Algebra
702(62)
Introduction
702(2)
Tensors
704(6)
Transformation of Tensor Coordinates
710(2)
Operations on Tensors
712(4)
Antisymmetric Tensors
716(14)
Tensors of First and Second Rank
730(9)
Properties of Dyads
739(13)
Tensor Mappings
752(12)
Tensor Analysis
764(77)
Introduction
764(2)
Point Spaces
766(2)
Rectilinear Coordinates
768(2)
Derivatives with Respect to Global Coordinates
770(5)
Curvilinear Coordinates
775(6)
Christoffel Symbols
781(6)
Derivatives with Respect to Local Coordinates
787(9)
Tensor Integrals
796(10)
Field Operations
806(13)
Nabla Calculus
819(6)
Special Vector Fields
825(4)
Integral Theorems
829(12)
Stochastics
Introduction
841(2)
Random Events
843(15)
Introduction
843(1)
Elementary Combinations
843(3)
Algebra of Events
846(2)
Probability
848(5)
Reliability
853(5)
Random Variables
858(48)
Introduction
858(2)
Probability Distributions
860(5)
Moments
865(3)
Functions of One Random Variable
868(4)
Functions of Several Random Variables
872(6)
Discrete Distributions
878(1)
Bernoulli Distribution
878(1)
Binomial Distribution
879(2)
Pascal Distribution
881(3)
Poisson Distribution
884(3)
Continuous Distributions
887(1)
Gamma Distribution
887(3)
Normal Distribution
890(5)
Logarithmic Normal Distribution
895(3)
Maximum Distributions
898(7)
Minimum Distributions
905(1)
Random Vectors
906(16)
Introduction
906(1)
Probability Distributions
907(5)
Moments
912(4)
Functions of a Random Vector
916(2)
Multinomial Distribution
918(2)
Multinormal Distribution
920(2)
Random Processes
922(27)
Introduction
922(4)
Finite Markov Processes in Discrete Time
926(1)
Introduction
926(1)
States and Transitions
926(6)
Structural Analysis
932(4)
Spectral Analysis
936(4)
First Passage
940(8)
Processes of Higher Order
948(1)
Finite Markov Processes in Continuous Time
949(27)
Introduction
949(1)
States and Transition Rates
949(5)
First Passage
954(5)
Queues
959(11)
Queue Systems
970(6)
Stationary Processes
976(15)
Introduction
976(1)
Probability Distributions and Moments
976(3)
Stationary Processes in Discrete Time
979(7)
Stationary Processes in Continuous Time
986(5)
Index 991

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