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.

This book is written as a guide for the presentation of experimental data including a consistent treatment of experimental errors and inaccuracies. It is meant for experimentalists in physics, astronomy, chemistry, life sciences and engineering. However, it can be equally useful for theoreticians who produce simulation data: they are often confronted with statistical data analysis for which the same methods apply as for the analysis of experimental data. The emphasis in this book is on the determination of best estimates for the values and inaccuracies of parameters in a theory, given experimental data. This is the problem area encountered by most physical scientists and engineers. The problem area of experimental design and hypothesis testing – excellently covered by many textbooks – is only touched on but not treated in this book.

The text can be used in education on error analysis, either in conjunction with experimental classes or in separate courses on data analysis and presentation. It is written in such a way – by including examples and exercises – that most students will be able to acquire the necessary knowledge from self study as well. The book is also meant to be kept for later reference in practical applications. For this purpose a set of “data sheets” and a number of useful computer programs are included.

This book consists of parts. Part I contains the main body of the text.

This chapter is about the presentation of experimental results. When the value of a physical quantity is reported, the uncertainty in the value must be properly reported too, and it must be clear to the reader what kind of uncertainty is meant and how it has been estimated. Given the uncertainty, the value must be reported with the proper number of digits. But the quantity also has a unit that must be reported according to international standards. Thus this chapter is about reporting your results: this is the last thing you do, but we'll make it the first chapter before more serious matters require attention.

How to report a series of measurements

In most cases you derive a result on the basis of a series of (similar) measurements. In general you do not report all individual outcomes of the measurements, but you report the best estimates of the quantity you wish to “measure,” based on the experimental data and on the model you use to derive the required quantity from the data. In fact, you use a data reduction method. In a publication you are required to be explicit about the method used to derive the end result from the data. However, in certain cases you may also choose to report details of the data themselves (preferably in an appendix or deposited as “additional material”); this enables the reader to check your results or apply alternative data reduction methods.

This appendix contains programs, functions or code fragments written in Python. Each code is referred to in the text; the page where the reference is made is given in the header.

First some general instructions are given on how to work with these codes. Python is a general-purpose interpretative language, for which interpreters are available for most platforms, including Windows. Python is in the public domain and interpreters are freely available. Most applications in this book use a powerful numerical array extension NumPy, which also provides basic tools in linear algebra, Fourier transforms and random numbers. Although Python version 3 is available, at the time of writing NumPy requires Python version 2, the latest being 2.6. In addition, applications may require the scientific tools library SciPy, which relies on NumPy. Importing SciPy automatically implies the import of NumPy.

Users are advised first to download Python 2.6, then the most recent stable version of NumPy, and then SciPy. Further instructions for Windows users can be found at www.hjcb.nl/python.

There are several options to produce plots, for example Gnuplot.py, based on the gnuplot package or rpy based on the statistical package “R.” But there are many more. Since the user may find it difficult to make a choice, we have added yet another, but very simple to use, plotting module called plotsvg.py. It can be downloaded from the author's website.