The vector concepts and techniques described in the previous chapters are important for two reasons: they allow you to solve a wide range of problems in physics and engineering, and they provide a foundation on which you can build an understanding of tensors (the “facts of the universe”). To achieve that understanding, you'll have to move beyond the simple definition of vectors as objects with magnitude and direction. Instead, you'll have to think of vectors as objects with components that transform between coordinate systems in specific and predictable ways. It's also important for you to realize that vectors can have more than one kind of component, and that those different types of component are defined by their behavior under coordinate transformations.

So this chapter is largely about the different types of vector component, and those components will be a lot easier to understand if you have a solid foundation in the mathematics of coordinate-system transformation.

Coordinate-system transformations

In taking the step from vectors to tensors, a good place to begin is to consider this question: “What happens to a vector when you change the coordinate system in which you're representing that vector?” The short answer is that nothing at all happens to the vector itself, but the vector's components may be different in the new coordinate system. The purpose of this section is to help you understand how those components change.

The real value of understanding vectors and how to manipulate them becomes clear when you realize that your knowledge allows you to solve a variety of problems that would be much more difficult without vectors. In this chapter, you'll find detailed explanations of four such problems: a mass sliding down an inclined plane, an object moving along a curved path, a charged particle in an electric field, and a charged particle in a magnetic field. To solve these problems, you'll need many of the vector concepts and operations described in Chapters 1 and 2.

Mass on an inclined plane

Consider the delivery woman pushing a heavy box up the ramp to her delivery truck, as illustrated in Figure 3.1. In this situation, there are a number of forces acting on the box, so if you want to determine how the box will move, you need to know how to work with vectors. Specifically, to solve problems such as this, you can use vector addition to find the total force acting on the box, and then you can use Newton's Second Law to relate that total force to the acceleration of the box.

To understand how this works, imagine that the delivery woman slips off the side of the ramp, leaving the box free to slide down the ramp under the influence of gravity.

It is normal practice when starting the mathematical investigation of a physical problem to assign algebraic symbols to the quantity or quantities whose values are sought, either numerically or as explicit algebraic expressions. For the sake of definiteness, in this chapter, our discussion will be in terms of a single quantity, which we will denote by x most of the time. The extension to two or more quantities is straightforward in principle, but usually entails much longer calculations, or a significant increase in complexity when graphical methods are used.

Once the sought-for quantity x has been identified and named, subsequent steps in the analysis involve applying a combination of known laws, consistency conditions and (possibly) given constraints to derive one or more equations satisfied by x. These equations may take many forms, ranging from a simple polynomial equation to, say, a partial differential equation with several boundary conditions. Some of the more complicated possibilities are treated in the later chapters of this book, but for the present we will be concerned with techniques for the solution of relatively straightforward algebraic equations.

When algebraic equations are to be solved, it is nearly always useful to be able to make plots showing how the functions, fi (x), involved in the problem change as their argument x is varied; here i is simply a label that identifies which particular function is being considered.

The qualitative approach

Ninety percent of all physics is concerned with vibrations and waves of one sort or another. The same basic thread runs through most branches of physical science, from acoustics through engineering, fluid mechanics, optics, electromagnetic theory and X-rays to quantum mechanics and information theory. It is closely bound to the idea of a signal and its spectrum. To take a simple example: imagine an experiment in which a musician plays a steady note on a trumpet or a violin, and a microphone produces a voltage proportional to the instantaneous air pressure. An oscilloscope will display a graph of pressure against time, F(t), which is periodic. The reciprocal of the period is the frequency of the note, 440 Hz, say, for a well-tempered middle A – the tuning-up frequency for an orchestra.

The waveform is not a pure sinusoid, and it would be boring and colourless if it were. It contains ‘harmonics’ or ‘overtones’: multiples of the fundamental frequency, with various amplitudes and in various phases, depending on the timbre of the note, the type of instrument being played and on the player. The waveform can be analysed to find the amplitudes of the overtones, and a list can be made of the amplitudes and phases of the sinusoids which it comprises. Alternatively a graph, A(ν), can be plotted (the sound-spectrum) of the amplitudes against frequency (Fig. 1.1).

In Chapter 6 we discussed how complicated functions f(x) may be expressed as power series. Although they were not presented as such, the power series could all be viewed as linear superpositions of the monomial basic set of functions, namely 1, x, x 2, x 3, … xn , … Natural though this set may seem, they are in many ways far from ideal: for example they possess no mutual orthogonality properties, a characteristic that is generally of great value when it comes to determining, for any particular function, the multiplying constant for each basic function in the sum. Moreover, this particular set of basic functions can only be used to represent continuous functions.

In the case of original functions f(t) that are periodic, some improvement on this situation can be made by using, as the basic set, sine and cosine functions. For a function with period T, say, the set of sine and cosine functions with arguments 2πnt/T, for all n ≥ 0, form a suitable basic set for expressing f as a series; such a representation is called a Fourier series. One great advantage they possess over the monomial functions is that they are mutually orthogonal when integrated over any continuous period of length T, i.e. the integral from t 0 to t 0 + T of the product of any sine and any cosine, or of two sines or cosines with different values of n, is equal to zero.

This and the next chapter are concerned with the formalism of probably the most widely used mathematical technique in the physical sciences, namely the calculus. The current chapter deals with the process of differentiation whilst Chapter 4 is concerned with its inverse process, integration. The topics covered are essential for the remainder of the book; once studied, the contents of the two chapters serve as reference material, should that be needed. Readers who have had previous experience of differentiation and integration should ensure full familiarity by looking at the worked examples in the main text and by attempting the problems at the ends of the two chapters.

Also included in this chapter is a section on curve sketching. Most of the mathematics needed as background to this important skill for applied physical scientists was covered in the first two chapters, but delaying our main discussion of it until the end of this chapter allows the location and characterisation of turning points to be included amongst the techniques available.

Differentiation

Differentiation is the process of determining how quickly or slowly a function varies, as the quantity on which it depends, its argument, is changed. More specifically, it is the procedure for obtaining an expression (numerical or algebraic) for the rate of change of the function with respect to its argument.

This chapter introduces space vectors and their manipulation. Firstly we deal with the description and algebra of vectors, then we consider how vectors may be used to describe lines, planes and spheres, and finally we look at the practical use of vectors in finding distances. The calculus of vectors will be developed in a later chapter; this chapter gives only some basic rules.

Scalars and vectors

The simplest kind of physical quantity is one that can be completely specified by its magnitude, a single number, together with the units in which it is measured. Such a quantity is called a scalar, and examples include temperature, time and density.

A vector is a quantity that requires both a magnitude (≥ 0) and a direction in space to specify it completely; we may think of it as an arrow in space. A familiar example is force, which has a magnitude (strength) measured in newtons and a direction of application. The large number of vectors that are used to describe the physical world include velocity, displacement, momentum and electric field. Vectors can also be used to describe quantities such as angular momentum and surface elements (a surface element has a magnitude, defined by its area, and a direction defined by the normal to its tangent plane); in such cases their definitions may seem somewhat arbitrary (though in fact they are standard) and not as physically intuitive as for vectors such as force.