Mathematical Foundations of Computational Engineering: A Handbook [Paperback]

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.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.- Numberló'