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Introduction

This unit shows how partial differential equations can be used to model phenomena such as waves and heat transfer. The prerequisite requirements to gain full advantage from this unit are an understanding of ordinary differential equations and basic familiarity with partial differential equations.

This unit is an adapted extract from the course Mathematical methods and models (MST209
Author(s): The Open University

1.5 Exercises

Exercise 1

A vector a has magnitude |a| = 7 and direction θ = −70°. Calculate the component form of a, giving the components correct to two decimal places.

<
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Learning outcomes

After studying this unit you should:

  • be able to perform basic algebraic manipulation with complex numbers;

  • understand the geometric interpretation of complex numbers;

  • know methods of finding the nth roots of complex numbers and the solutions of simple polynomial equations.


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Introduction

This unit is an adapted extract from the Open University course Complex analysis (M337)

This unit is devoted solely to complex numbers.

In Section 1, we define complex numbers and show you how to manipulate them, stressing the similarities with the manipulation of real numbers.

Section 2 is devoted to the geometric representation of complex numbers. You will find that
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1.3.5 Journals

Journals and articles written by academics or experts are an excellent source of information. Journals are usually published monthly or quarterly, and contain a selection of articles providing details of recent research. Often they will also contain reviews of relevant books. They are usually published more quickly than books, and so are often more up to date.

To access content of journals, most publishers require a subscription. There are, however, some journals which you can freely ac
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1.2.2 Choosing keywords

Keywords are significant words which define the subject you are looking for. The importance of keywords is illustrated by the fact that there is a whole industry around providing advice to companies on how to select keywords for their websites that are likely to make it to the top of results lists generated by search engines. We often choose keywords as part of an iterative process; usually if we don't hit on the right search terms straight off, most of us tweak them as we go along based on t
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5 Conclusion – new ways of looking at the world

There is a variety of new approaches or terms that are interlinked, and have been prominent throughout this book. All of them have played a part in this book's journey through the scientific, political, philosophical and social implications of climate change.

Governance of climate change is about: decision making under uncertainty; understanding and representing vulnerability even when vulnerabilities are difficult to assess or unknowable; and making every aspect of human
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Learning outcomes

By the end of this unit you should be able to:

  • appreciate how chemical processes in the rest of the world affect the Arctic environment and the species inhabiting it;

  • recognise the physical processes that determine atmosphere and oceanic flows in the Arctic;

  • appreciate the scientific research process and the use of scientific evidence;

  • use quantitative scientific evidence to examine the link between atmospheric carbon dioxide levels a
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6.3 The role of active citizens and communities

Few people agree that individuals should take the main responsibility for tackling environmental issues. For example, in a 2007 poll of over 2000 UK citizens, 70% agreed that the government should take a lead in combating climate change, even if it means using the law to change people's behaviour. However, over 60% disagreed that there was nothing they could do to avert climate change and over half agreed that they would do more if others did more too, although 40% thought that recycling was
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1.1.1 Carbon dioxide and other greenhouse gases

Carbon dioxide (CO2) is used as the basis for the carbon footprint because it is by far the main contributor to the enhanced greenhouse effect from human activity (mainly burning fossil fuels, clearing forests and making cement). So, often only CO2 is counted in the carbon footprint. However, for a more complete measure of the carbon footprint the other human-generated greenhouse gases are converted into a CO2 equivalent (in kilograms or tonnes CO2e
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1.1 What is the carbon footprint and why is it important?

The carbon footprint is the annual amount of greenhouse gas emissions, mainly carbon dioxide, that result from the activities of an individual or a group of people, especially their use of energy and transport and consumption of goods and services. It's measured as the mass, in kilograms or tonnes per year, either of carbon dioxide (CO2) emissions alone, or of the carbon dioxide equivalent (CO2e) effect of other greenhouse gas emissions.


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2.6.2 The role of modelling studies

State-of-the-art models are designed to simulate the workings of the climate system (in so far as this is currently understood), and include the ‘internal’ interactions that generate short-term natural variability in the real world. They provide modellers with a means of carrying out ‘virtual’ experiments on the climate system. In the present context, an important aim of these experiments is to identify the ‘signal’ of a human influence on climate, so studies typically involve ‘
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2.6.1 Weighing up the evidence: the full cast of suspects

Figure 36 (again adapted from the TAR) takes your thoughts on Question 11 on a stage. It gives estimates of the cumulative effect since pre-industrial times of the various climate change agents, with the contributions expressed in terms of radiative forcing. Note that the figure also includes yet another device for communicating the IPCC's confidence in a particular finding – an indication of the ‘level of scientific understanding’ that accompanies each estimate. This reflects the autho
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2.5.2 Environmental indicators

The notion of a link between climatic conditions and the behaviour of plants and animals (e.g. the growth of trees or coral) and the composition of natural communities or ecosystems (the type of vegetation in a given area, say) is fundamental to the use of proxy data to reconstruct past climates. Some examples of biological responses to recent climate change were included in Box 9. Here we should be wary of jumping to conclusions. Such changes involve complex living systems that can respond i
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2.5.1 Physical and weather-related indicators

The indicators collected in Table 4 have been observed to change over large regions of the Earth during the 20th century. According to the TAR, there is now a good level of confidence that what is being recorded is the result of long-term change rather than short-term natural fluctuations. As we noted earlier (Section 2.2.2), the most recent period of warming has been almost global in extent, but particularly marked at high latitudes. So are the changes in Table 4 consistent with rising tempe
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References

Capra, F. (1996) The Web of Life. Used by permission of Doubleday, a division of Random House, Inc., and, in the UK, reprinted by permission of HarperCollins Publishers Ltd.
Capra, F. (2002) The Hidden Connections: Integrating the Biological, Cognitive, and Social Dimension of Life into a Science of Sustainability. Used by permission of Doubleday, a division of Random House, Inc., and, in the UK, reprin
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Learning outcomes

By the end of this unit you should be able to:

  • understand why systems thinking might be useful and know something about how it can be applied in the context of environmental responsibility;

  • describe the significance of environmental pragmatism and cognitive justice as tools for supporting environmental policy and action.


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1.2.8 In praise of cheap offshore labour? continued

There are two points which are central to this line of thinking. One, according to Wolf (2004), is that the whole process, as odd as it may sound, is about mutual exploitation. Outside firms do indeed exploit the poor by taking advantage of the profitable opportunities that a pool of cheap labour represents. But Indonesian or Chinese workers, for instance, could be said to exploit the incoming firms by extracting higher pay from them and taking advantage of opportunities that previousl
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1.2.7 In praise of cheap offshore labour?

Claims over the benefits of globalisation and the exploitation of cheap offshore labour generate strong feelings and, not surprisingly, divide opinion between those who favour the global marketplace and its detractors. The issue turns on whether the constant search for ever-cheaper manufacturing and service locations is seen as a good or a bad thing. It may appear odd, at first, to suggest that exploiting the poor of another country can, on any measure, be regarded as a good thing, but
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1.1.2 Activity 1

You have already glanced at Figure 1 and some of the worki
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