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In Chap. 14 it was shown how a matrix A and a vector r can be multiplied to give a new vector r' (provided the dimensions of the vector and the matrix fit): r' = Ar.

Consider a steady flow of water through a pipe. The water is assumed to be incompressible, i.e. it has a uniform density (for which we will use the symbol ρ), the velocity of each particle having a constant value v = ds/dt.

In the preceding chapter ″Fourier series″ we showed that an arbitrary periodic function with period T can be described as the sum of trigonometric functions with multiples of the period T.

The purpose of this chapter is to consider some of the important applications of integration as applied to problems in physics and engineering. Its objective is twofold. Firstly, it demonstrates the practical use of the integral calculus to readers who are particularly interested in applications. Secondly, other readers may use this chapter as a reference when practical problems are encountered.

We now ask whether a function can be expanded in terms of functions other than power functions, and especially whether a periodic function may be expanded in terms of periodic functions, say trigonometric functions.Many problems in physics and engineering involve periodic functions, particularly in electrical engineering, vibrations, sound and heat conduction.

In Chap. 5 we showed that the sum of a geometric series is given by $$1+x+x^{2}+x^{3}+\cdots=\frac{1}{1-x}$$ This formula holds true for \(-1

One important aspect in the solution of physical and engineering problems is the choice of coordinate systems. The right choice may considerably reduce the degree of difficulty and the length of the necessary computations.

In this chapter we will investigate the solution of sets of linear algebraic equations. First, we show a method which will be used in most practical cases. This is the Gaussian method of elimination and its refinements. The basic idea is quite clear and elementary. Notation in matrix form will prove to be helpful.

Mathematics is the basic language in physics and engineering. It is an essential tool which first and second year students have to master as soon as possible. A lack of competence in mathematics is the main reason for failure and drop out in the beginning periods of study. This textbook offers an accessible and highly approved approach which is characterized by the combination of the textbook with a detailed study guide available online at extras.springer.com. This study guide divides the whole learning task into small units which the student is very likely to master successfully. Thus he or she is asked to read and study a limited section of the textbook and to return to the study guide afterwards. Working with the study guide his or her learning results are controlled, monitored and deepened by graded questions, exercises, repetitions and finally by problems and applications of the content studied. Since the degree of difficulties is slowly rising the students gain confidence and experience their own progress in mathematical competence thus fostering motivation. Furthermore in case of learning difficulties he or she is given supplementary explanations and in case of individual needs supplementary exercises and applications. So the sequence of the studies is individualised according to the individual performance and needs and can be regarded as full tutorial course. More than that the study guide aims to satisfy two objectives simultaneously: firstly it enables students to make effective use of the textbook and secondly it offers advice on the improvement of study skills. Empirical studies have shown that the student's competence for using written information has improved significantly by using this study guide.

So far we have dealt with functions of only a single variable, such as x, t etc. But functions of more than one variable also occur frequently in physics and engineering.

Mathematics is used in physics and engineering to describe natural events in which quantities are specified by numerical values and units of measurement. Such a description does not always lead to a successful conclusion. Consider, for example, the following statement from a weather forecast: ‘There is a force 4 wind over the North Sea.’ In this case we do not know the direction of the wind, which might be important. The following forecast is complete: ‘There is a force 4 westerly wind over the North Sea.’ This statement contains two pieces of information about the air movement, namely the wind force which would be measured in physics as a wind velocity in metres per second (m/s) and its direction. If the direction was not known, the movement of the air would not be completely specified.Weather charts indicate the wind direction by means of arrows, as shown in Fig. 1.1. It is evident that this is of considerable importance to navigation. The velocity is thus completely defined only when both its direction and magnitude are given. In physics and engineering there are many quantities which must be specified by magnitude and direction. Such quantities, of which velocity is one, are called vector quantities or, more simply, vectors.

Let us develop the problem by a simple example. A solid cube, as shown in Fig. 13.1, has a volume V . If the density p is constant throughout the entire volume then the mass is given by M = pV There are cases, however, in which the density p is not constant throughout the volume. The density of the Earth is greater near the center than at the surface. The density of the atmosphere is at a maximum at the surface of the Earth: it decreases exponentially with the altitude.

The theory of errors is a part of mathematical statistics and deals with the following facts. Given the results of measurements carried out in a laboratory, we require statements about the ‘true’ value of the measured quantity and a prediction of the accuracy of the measurements. There are two types of errors which arise when we carry out a measurement: systematic or constant errors and random errors.

Consider the multiplication of a number by itself, for example a×a×a. A simple way of expressing this is to write a3. If we multiplied a by itself n times we would write an. We would say a to the power n;n is known as an index or exponent.

Seit Beginn des CIP-Programms werden an den CIP-Gerate-Clustern des Hochschul-rechenzentrums der Universitat Frankfurt am Main auch Weiterbildungs-veranstaltungen fur das wissenschaftliche und nicht-wissenschaftliche Personal koordiniert und durchgefuhrt. Die Inhalte dieser Veranstaltungen sind zumeist Einfuhrungen in die Anwendung kommerzieller PC-Standardsoftware (Textverarbeitung, Desk-Top-Publishing, Datenbankeinsatz, Grafik) sowie Einfuhrungen in den Umgang mit dem Betriebssystem MS-DOS und den Geraten selbst.

Ziel der Software-Ergonomie ist eine benutzergerechte Gestaltung von Computerssystemen [4]. Als Minimum ist von ergonomisch gestalteter Software zu fordern, das dem Benutzer einfach zu benutzende Befehle (oder allgemeiner: Manipulationsmodi) zur Verfugung stehen, die auf einer transparenten und konsistenten Syntax aufbauen. Fur naive Anwender und fur Anfanger wird angenommen, das die Auswahl gegebener Befehle aus einem Menu einer kommandoorientierten Bedienungsweise vorzuziehen ist; einschlagige Untersuchungen stutzen diese Annahme [5].

Zum strukturierten Entwurf von Programmen in imperativen Programmiersprachen dienen allgemeine Konzepte, die unabhangig von der speziellen Programmiersprache sind. In den einzelnen Programmiersprachen sind sie in verschiedener Form und Auswahl realisiert. Sie werden deshalb in diesem Kapitel als Grundlage fur die Ausfuhrungen in den Kapiteln III bis VII bereitgestellt.

Funktionale Programmierung bedeutet, das Programme aus einer Menge von Funktionsdefinitionen und Funktionsanwendungen bestehen. Mit funktionalen Sprachen sollen vornehmlich Aufgaben der Wissensdarstellung und Wissensverarbeitung (symbolische Datenverarbeitung) und nicht in erster Linie numerische oder verwandte Problemstellungen bearbeitet werden. Die wichtigste Datenstruktur ist die Liste, d.h. eine endliche, aber beliebig lange Folge verschiedener Datenelemente. Auch die Texte von Funktionsdefinitionen werden als Liste notiert, ebenso die Aufrufe von Funktionen.

Dem logischen Programmieren liegt der Gedanke zugrunde, das fur ein Problem bedeutsame Wissen als Folge von Aussagen und von Regeln (sog. Klauseln) zu formulieren und anschliesend eine Frage zu stellen, die vom Computersystem als ein zu beweisendes Theorem behandelt wird. Prolog als Hauptvertreter fur dieses Konzept wird daher als eine deklarative Sprache bezeichnet. Man programmiert keinen Algorithmus; diese neue Form des Programmierens weist zunachst gar keine imperativen (prozeduralen) Elemente auf. Die Strategie zum Theorembeweis ist in Prolog fest vorgegeben.