Mathematical Foundations of Computational Engineering de Peter J. Pahl

Mathematical Foundations of Computational Engineering: A Handbook

Peter J. Pahl

F. Pahl

Rudolf Damrath

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.

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Cuprins

Logic.- 1.1 Representation of Thought.- 1.2 Elementary Concepts.- 1.3 Propositional Logic.- 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.- 3.5 Vector Spaces.- 3.6 Linear Mappings.- 3.7 Vector and Matrix Algebra.- 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.- 5.8 Connectedness of Sets.- 5.9 Separation Properties.- 5.10 Convergence.- 5.11 Compactness.- 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.- 7.2 Groups and Subgroups.- 7.3 Types of Groups.- 7.4 Class Structure.- 7.5 Group Structure.- 7.6 Abelian Groups.- 7.7 Permutations.- 7.8 General Groups.- 7.9 Unique Decomposition of Abelian Groups.- Graphs.- 8.1 Introduction.- 8.2 Algebra of Relations.- 8.3 Classification of Graphs.- 8.4 Structure of Graphs.- 8.5 Paths in Networks.- 8.6 Network Flows.- Tensors.- 9.1 Introduction.- 9.2 Vector Algebra.- 9.3 Tensor Algebra.- 9.4 Tensor Analysis.- Stochastics.- 10.1 Introduction.- 10.2 Random Events.- 10.3 Random Variables.- 10.4 Random Vectors.- 10.5 Random Processes.

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o.Prof. Dr. Dr. h.c. mult. Peter Jan Pahl is an expert in the field of computational engineering, with international activities. He is proposed President of the International Society on Computing in Civil and Structural Engineering. Prof. Dr.-Ing. Rudolf Damrath worked with Professor Pahl for over 20 years and is now international renown himself. They both worked on fundamental topics in computational engineering, especially for numerical applications in civil and structural engineering. This book serves as a comprehensive handbook, bringing the fundamental mathematics of engineering computation into a suitable order for specific engineering purposes, showing their siginificance for typical applications. The relevant definitions, notations and theories are presented in a durable form which is independent of the fast development of information technology.

Caracteristici

Gives explanations and definitions of the basic concepts of modern computational engineering Bundles information which could not be found collected in a single book before Brings together the basic concepts into a general approach to computational models for engineering tasks Includes supplementary material: sn.pub/extras