基本説明
Contents: Mathematical Logic. -Sets. -Algebraic Structures. -Ordinal Structures. -Topology. -Number System. -Groups. -Graphs. -Tensors. -Stochastics.
Full Description
Computational engineering is the treatment of engineering tasks with computers. It is based on computational mathematics, which is presented here in a comprehensive handbook. From the existing rich repertoire of mathematical theories and methods, the fundamentals of engineering computation are here presented in a coherent fashion. They are brought into a suitable order for specific engineering purposes, and their significance for typical applications 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.
Contents
Logic.- 1.1 Representation of Thought.- 1.2 Elementary Concepts.- 1.3 Propositional Logic.- 1.3.1 Logical Variables and Connectives.- 1.3.2 Logical Expressions.- 1.3.3 Logical Normal Form.- 1.3.4 Logical Rules of Inference.- 1.4 Predicate Logic.- 1.5 Proofs and Axioms.- Set Theory.- 2.1 Sets.- 2.2 Algebra of Sets.- 2.3 Relations.- 2.4 Types of Relations.- 2.5 Mappings.- 2.6 Types of Mappings.- 2.7 Cardinality and Countability.- 2.8 Structures.- Algebraic Structures.- 3.1 Introduction.- 3.2 Inner Operations.- 3.3 Sets with One Operation.- 3.4 Sets with Two Operations.- 3.4.1 Introduction.- 3.4.2 Additive and Multiplicative Domains.- 3.4.3 Dual Domains.- 3.5 Vector Spaces.- 3.5.1 General Vector Spaces.- 3.5.2 Real Vector Spaces.- 3.6 Linear Mappings.- 3.7 Vector and Matrix Algebra.- 3.7.1 Definitions.- 3.7.2 Elementary Vector Operations.- 3.7.3 Elementary Matrix Operations.- 3.7.4 Derived Scalars.- 3.7.5 Complex Vectors and Matrices.- Ordinal Structures.- 4.1 Introduction.- 4.2 Ordered Sets.- 4.3 Extreme Elements.- 4.4 Ordered Sets with Extremality Properties.- 4.5 Mappings of Ordered Sets.- 4.6 Properties of Ordered Sets.- 4.7 Ordered Cardinal Numbers.- Topological Structures.- 5.1 Introduction.- 5.2 Topological Spaces.- 5.3 Bases and Generating Sets.- 5.4 Metric Spaces.- 5.5 Point Sets in Topological Spaces.- 5.6 Topological Mappings.- 5.7 Construction of Topologies.- 5.7.1 Final and Initial Topologies.- 5.7.2 Subspaces.- 5.7.3 Product Spaces.- 5.8 Connectedness of Sets.- 5.8.1 Disconnections and Connectedness.- 5.8.2 Connectedness of Constructed Sets.- 5.8.3 Components and Paths.- 5.9 Separation Properties.- 5.10 Convergence.- 5.10.1 Sequences.- 5.10.2 Subsequences.- 5.10.3 Series.- 5.10.4 Nets.- 5.10.5 Filters.- 5.11 Compactness.- 5.11.1 Compact Spaces.- 5.11.2 Compact Metric Spaces.- 5.11.3 Locally Compact Spaces.- 5.12 Continuity of Real Functions.- Number System.- 6.1 Introduction.- 6.2 Natural Numbers.- 6.3 Integers.- 6.4 Rational Numbers.- 6.5 Real Numbers.- 6.6 Complex Numbers.- 6.7 Quaternions.- Groups.- 7.1 Introduction.- 7.1.1 Group Theory.- 7.1.2 Outline.- 7.2 Groups and Subgroups.- 7.3 Types of Groups.- 7.3.1 Permutation Groups.- 7.3.2 Symmetry Groups.- 7.3.3 Generated Groups.- 7.3.4 Cyclic Groups.- 7.3.5 Groups of Integers.- 7.3.6 Cyclic Subgroups.- 7.4 Class Structure.- 7.4.1 Classes.- 7.4.2 Cosets and Normal Subgroups.- 7.4.3 Groups of Residue Classes.- 7.4.4 Conjugate Elements and Sets.- 7.5 Group Structure.- 7.5.1 Introduction.- 7.5.2 Homomorphism.- 7.5.3 Isomorphism.- 7.5.4 Isomorphic Types of Groups.- 7.5.5 Automorphisms.- 7.6 Abelian Groups.- 7.6.1 Introduction.- 7.6.2 Classification of Abelian Groups.- 7.6.3 Linear Combinations.- 7.6.4 Direct Sums.- 7.6.5 Constructions of Abelian Groups.- 7.6.6 Decompositions of Abelian Groups.- 7.7 Permutations.- 7.7.1 Introduction.- 7.7.2 Symmetric Groups.- 7.7.3 Cycles.- 7.7.4 Conjugate Permutations.- 7.7.5 Transpositions.- 7.7.6 Subgroups of a Symmetric Group.- 7.7.7 Group Structure of the Symmetric Group S4.- 7.7.8 Class Structure of the Symmetric Group S4.- 7.8 General Groups.- 7.8.1 Introduction.- 7.8.2 Classes in General Groups.- 7.8.3 Groups of Prime-power Order.- 7.8.4 Normal Series.- 7.9 Unique Decomposition of Abelian Groups.- Graphs.- 8.1 Introduction.- 8.2 Algebra of Relations.- 8.2.1 Introduction.- 8.2.2 Unary Relations.- 8.2.3 Homogeneous Binary Relations.- 8.2.4 Heterogeneous Binary Relations.- 8.2.5 Unary and Binary Relations.- 8.2.6 Closures.- 8.3 Classification of Graphs.- 8.3.1 Introduction.- 8.3.2 Directed Graphs.- 8.3.3 Bipartite Graphs.- 8.3.4 Multigraphs.- 8.3.5 Hypergraphs.- 8.4 Structure of Graphs.- 8.4.1 Introduction.- 8.4.2 Paths and Cycles in Directed Graphs.- 8.4.3 Connectedness of Directed Graphs.- 8.4.4 Cuts in Directed Graphs.- 8.4.5 Paths and Cycles in Simple Graphs.- 8.4.6 Connectedness of Simple Graphs.- 8.4.7 Cuts in Simple Graphs.- 8.4.8 Acyclic Graphs.- 8.4.9 Rooted Graphs and Rooted Trees.- 8.5 Paths in Networks.- 8.5.1 Introduction.- 8.5.2 Path Algebra.- 8.5.3 Boolean Path Algebra.- 8.5.4 Real Path Algebra.- 8.5.4.1 Minimal Path Length.- 8.5.4.2 Maximal Path Length.- 8.5.4.3 Maximal Path Reliability.- 8.5.4.4 Maximal Path Capacity.- 8.5.5 Literal Path Algebra.- 8.5.5.1 Path Edges.- 8.5.5.2 Common Path Edges.- 8.5.5.3 Simple Paths.- 8.5.5.4 Extreme Simple Paths.- 8.5.5.5 Literal Vertex Labels.- 8.5.5.6 Literal Edge Labels for Simple Graphs.- 8.5.5.7 Applications in Structural Analysis.- 8.5.6 Properties of Path Algebras.- 8.5.7 Systems of Equations.- 8.5.7.1 Solutions of Systems of Equations.- 8.5.7.2 Direct Methods of Solution.- 8.5.7.3 Iterative Methods of Solution.- 8.6 Network Flows.- 8.6.1 Introduction.- 8.6.2 Networks and Flows.- 8.6.3 Unrestricted Flow.- 8.6.4 Restricted Flow.- 8.6.5 Maximal Flow.- 8.6.6 Maximal Flow and Minimal Cost.- 8.6.7 Circulation.- Tensors.- 9.1 Introduction.- 9.2 Vector Algebra.- 9.2.1 Vector Spaces.- 9.2.2 Bases.- 9.2.3 Coordinates.- 9.2.4 Metrics.- 9.2.5 Construction of Bases.- 9.2.6 Transformation of Bases.- 9.2.7 Orientation and Volume.- 9.3 Tensor Algebra.- 9.3.1 Introduction.- 9.3.2 Tensors.- 9.3.3 Transformation of Tensor Coordinates.- 9.3.4 Operations on Tensors.- 9.3.5 Antisymmetric Tensors.- 9.3.6 Tensors of First and Second Rank.- 9.3.7 Properties of Dyads.- 9.3.8 Tensor Mappings.- 9.4 Tensor Analysis.- 9.4.1 Introduction.- 9.4.2 Point Spaces.- 9.4.3 Rectilinear Coordinates.- 9.4.4 Derivatives with Respect to Global Coordinates.- 9.4.5 Curvilinear Coordinates.- 9.4.6 Christoffel Symbols.- 9.4.7 Derivatives with Respect to Local Coordinates.- 9.4.8 Tensor Integrals.- 9.4.9 Field Operations.- 9.4.10 Nabla Calculus.- 9.4.11 Special Vector Fields.- 9.4.12 Integral Theorems.- Stochastics.- 10.1 Introduction.- 10.2 Random Events.- 10.2.1 Introduction.- 10.2.2 Elementary Combinatorics.- 10.2.3 Algebra of Events.- 10.2.4 Probability.- 10.2.5 Reliability.- 10.3 Random Variables.- 10.3.1 Introduction.- 10.3.2 Probability Distributions.- 10.3.3 Moments.- 10.3.4 Functions of One Random Variable.- 10.3.5 Functions of Several Random Variables.- 10.3.6 Discrete Distributions.- 10.3.6.1 Bernoulli Distribution.- 10.3.6.2 Binomial Distribution.- 10.3.6.3 Pascal Distribution.- 10.3.6.4 Poisson Distribution.- 10.3.7 Continuous Distributions.- 10.3.7.1 Gamma Distribution.- 10.3.7.2 Normal Distribution.- 10.3.7.3 Logarithmic Normal Distribution.- 10.3.7.4 Maximum Distributions.- 10.3.7.5 Minimum Distributions.- 10.4 Random Vectors.- 10.4.1 Introduction.- 10.4.2 Probability Distributions.- 10.4.3 Moments.- 10.4.4 Functions of a Random Vector.- 10.4.5 Multinomial Distribution.- 10.4.6 Multinormal Distribution.- 10.5 Random Processes.- 10.5.1 Introduction.- 10.5.2 Finite Markov Processes in Discrete Time.- 10.5.2.1 Introduction.- 10.5.2.2 States and Transitions.- 10.5.2.3 Structural Analysis.- 10.5.2.4 Spectral Analysis.- 10.5.2.5 First Passage.- 10.5.2.6 Processes of Higher Order.- 10.5.3 Finite Markov Processes in Continuous Time.- 10.5.3.1 Introduction.- 10.5.3.2 States and Transition Rates.- 10.5.3.3 First Passage.- 10.5.3.4 Queues.- 10.5.3.5 Queue Systems.- 10.5.4 Stationary Processes.- 10.5.4.1 Introduction.- 10.5.4.2 Probability Distributions and Moments.- 10.5.4.3 Stationary Processes in Discrete Time.- 10.5.4.4 Stationary Processes in Continuous Time.
