Klaus Weltner

Theory of Errors

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.

Applications of Integration

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.

Fourier Integrals and Fourier Transforms

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.

Vector Analysis: Surface Integrals, Divergence, Curl and Potential

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.

Vector Algebra II: Scalar and Vector Products

We saw in the previous chapter how vector quantities may be added and subtracted. In this chapter we consider the products of vectors and define rules for them. First we will examine two cases frequently encountered in practice.

Fourier Series; Harmonic Analysis

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.

Functions of Several Variables; Partial Differentiation; and Total Differentiation

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.

Transformation of Coordinates; Matrices

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.

Vector Algebra I: Scalars and Vectors

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.

Multiple Integrals; Coordinate Systems

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.