Matematik for Ingeniører

PURPOSE. This textbook system (3 volumes in all) intends to give a short and easily read presentation of the mathematics, which is useful for an ingineering job or needed to understand courses and litterature about many technical and scientific subjects. The books should also serve as a dictionary and a source of inspiration for engineers and others with a need for applied mathematics. -- Each volume is used for 1 course: approximately 45 hours + homework. ---------- ---------- ---------- ---------- READER LEVEL. Mathematics at level A from the danish high school ("gymnasiet") is assumed. But the most important basics are recapitulated, such that readers with mathematics at level B should be able to catch up.---------- ---------- --------------------GENERAL FOR ALL 3 VOLUMES. Most topics are accompanied by numerical methods (and algorithms), which are useful for an engineer and helps the mathematical understanding by giving a realistic representation. It is further shown that mathematical software systems can be used to solve the problems. For these examples we have chosen the software package "Maple".-------------------- ---------- ------------------------------STRUCTURE OF ALL 3 VOLUMES. Each chapter starts with the most indispensable topics, which we try to explain in an intuitive and practical way by graphs and examples. Definitions and theorems are formulated precisely, and most of the theorems are proved. Later in the chapter there are some supplementary sections containing more peripheral topics, deeper explanations, and long complicated proofs. And at the end of each chapter are the exercises (with answers at the end of each volume). -- In appendix there are algorithms for the numerical methods, and surveys with useful formulas. ---------- ---------- ---------- CONTENT OF VOLUME 1. Basic concepts for function of 1 variable (much is known from high school). Differential equations. Complex numbers. Approximation. ---- ---------- ---------- ---------- -------------------- ---------- ---------- ----------DETAILED DANISH CONTENTS: ---------- ---------- -------------------- 1 FUNKTION AF EN VARIABEL (page 1) >>>>> 1.1 Funktionsbegrebet. 1.2 Graensevaerdi. 1.3 Kontinuitet. 1.4 Differentiation. 1.5 Integration. 1.6 Numerisk integration. 1A Praecis definition af "graensevaerdi". 1B Uniform kontinuitet. -- Opgaver. ---------- ---------- ---------- ---------- ---------- ---------- ----------2 STANDARDFUNKTIONER (page 39) >>>>> 2.1 Logaritmefunktioner. 2.2 Eksponentialfunktioner. 2.3 Potensfunktioner. 2.4 Logaritmiske koordinatsystemer. 2.5 Trigonometriske funktioner. 2.6 Arcusfunktioner. 2.7 De hyperbolske funktioner. 2.8 Areafunktionerne. 2A Detaljeret definition af potensfunktioner. 2B Bevis for trigonometriske formler. -- Opgaver.--------- --------- ------------------ --------- ---------3 GRAENSEUNDERSOEGELSER OG INTEGRATION (page 78) >>>>> 3.1 Indledning. 3.2 L'Hospitals regel. 3.3 Stoerrelsesforhold. 3.4 Uegentlige integraler. 3.5 Dekomposition. 3A Bevis for l'Hospitals regel. 3B Cauchy's hovedværdi. -- Opgaver.---------- ---------- -------------------- ---------- ---------- ----------4 DIFFERENTIALLIGNINGER AF 1. ORDEN >>>>> 4.1 Indledning. 4.2 Numerisk loesning af 1. ordens differentialligning. 4.3 Differentialligninger, hvor de variable kan separeres. 4.4 Den lineære differentialligning af foerste orden. 4A Separere variable. 4B Andre typer af differentialligninger af 1. orden. -- Opgaver.---------- ---------- ---------- ---------- ---------- ---------- 5 KOMPLEKSE TAL >>>>>5.1 Indledning. 5.2 Maengden af komplekse tal. 5.3 Addition og subtraktion. 5.4 Multiplikation og division. 5.5 Rektangulaer form. 5.6 Polaer form. Eksponentialfunktionen. 5.7 Den binome ligning. 5.8 Andengradsligningen. 5.9 Polynomier. 5A Algebraens fundamentalsætning. 5B Dekomposition i det generelle tilfælde. -- Opgaver.---------- ---------- ---------- ---------- ---------- ---------- ---------- 6 DIFFERENTIALLIGNINGSPROBLEM AF VILKAARLIG ORDEN >>>>> 6.1 Indledning. 6.2 Den homogene lineaere differentialligning af 2. orden med konstante koefficienter. 6.3 Den homogene lineaere differentialligning af n'te orden med konstante koefficienter. 6.4 Den inhomogene differentiallign ing med konstante koefficienter. 6.5 Den lineaere differentialligning af 2. orden. 6.6 System af 1. ordens differentialligninger. Numerisk loesning. 6.7 Omformning af differentialligninger af n'te orden. 6A Bevis vedr. lineaer differentialligning af 2. orden. -- Opgaver.---------- ---------- ---------- ---------- ---------- ---------- -------------------- ---------- ----------7 APPROKSIMERENDE FUNKTION (page 228) >>>>>7.1 Anvendelser. 7.2 Taylor-approksimation for funktion af 1 variabel. 7.3 Polynomie-kollokation.7.4 Spline-kollokation. 7A Mere om Taylorpolynomium. 7B Rational kollokation. 7C Benyttelse af tabel ved spline-kollokation. -- Opgaver.---------- ---------- ---------- ---------- ---------- ---------- -------------------- ----------APPENDIX >>>>> 1.1 Numerisk differentiation. 1.2 Numerisk integration. 2.1 Standardfunktioner. 4.1 Numerisk loesning af 1. ordens differentialligning. 4.2 Oversigt over differentialligninger af 1. orden. 5.1 Komplekse standardfunktioner. 5.2 Loesning af anden- og tredie gradsligning. 5.3 Loesning af n'te gradsligning. 6.1 Numerisk loesning af 2. ordens differentialligning. 6.2 Omregning af svingninger. 6.3 Gaettemetode. Partikulaer loesning til lineaer differentialligning. 6.4 Tabelmetode. Partikulaer loesning til lineaer differentialligning. 6.5 Integralmetode. Partikulaer loesning til lineaer differentialligning. 6.6 Numerisk loesning af system af differentialligninger. 6.7 Oversigt over differentialligninger af 2. orden. 7.1 Approksimation af f(x).---------- ---------- ---------- --------- LITTERATUR (page 289), FACITLISTE (page 292), SYMBOLLISTE (page 318), STIKORD (page 322).