This unit introduces the topic of differential equations. The subject is developed without assuming that you have come across it before, but it is taken for granted that you have a basic grounding in calculus. In particular, you will need to have a good grasp of the basic rules for differentiation and integration.

This unit is an adapted extract from the course Mathematical methods and
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Acknowledgements

All materials included in this unit are derived from content originated at the Open University.

Author(s): The Open University

1.3: Summing vectors given in geometric form

The following activity illustrates how the conversion processes outlined in the preceding sections may come in useful. If two vectors are given in geometric form, and their sum is sought in the same form, one approach is to convert each of the vectors into component form, add their corresponding components, and then convert the sum back to geometric form.

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1.2: Converting to geometric form

You have seen how any vector given in geometric form, in terms of magnitude and direction, can be written in component form. You will now see how conversion in the opposite sense may be achieved, starting from component form. In other words, given a vector aÂ =Â a 1 iÂ +Â a 2 j, what are its magnitude |a| and direction Î¸?

The first part of this question is dealt with using Pythagorasâ€™ Theorem: the magnitude of a v
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Acknowledgements

All materials included in this unit are derived from content originated at the Open University.

Author(s): The Open University

4.3.3 Pipe dreams?

The idea underlying complementary currencies â€“ that there is a great well of social capital waiting to be drawn upon to make society more sustainable â€“ is an idea that is becoming quietly influential. â€˜Social capitalâ€™ is a term frequently used by those mainstream politicians and civil servants tasked with addressing the widening gap between rich and poor people within societies throughout the world. Indeed, investing in and enhancing social capital is now the starting point in
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Acknowledgements

The material acknowledged below is Proprietary and used under licence, see terms and conditions). This content is made available under a Creative Commons Attribution-NonCommercial-ShareAlike 2.0 Licence

Grateful acknowledgement is made to the following:

## Figures

Figur
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1 A climate change icon

The polar bear has become an international climate change icon. But how much is known about this bear, its habitat and life? This unit will talk about the role of language, but by way of introduction how about the name of this bear? To me it is the polar bear; to a German it is an EisbÃ¤r (ice bear) and to a French person it is an ours blanc (white bear). In these three examples the bear is referred to as polar, white, or an ice bear â€“ eminently sensible. The Latin name for th
Author(s): The Open University

Introduction

The scientific theory of plate tectonics suggests that at least some of these Arctic lands were once tropical. Since then the continents have moved and ice has changed the landscape. This unit will concentrate on evidence from the last 800,000 years using information collected from ice cores from Greenland and Antarctica, and will use this evidence to discuss current and possible future climate. The cores show that there have been nine periods in the recent past when large areas of the Earth
Author(s): The Open University

6.1 ‘I’, ‘we’ or ‘they’?

We must abandon the conceit that individual, isolated, private actions are the answer. They can and do help. But they will not take us far enough without collective action.

(Al Gore, 2007)

There are some things that we can do as individuals: making this an energy-efficient house and making smart transport choices. Then there a
Author(s): The Open University

5.3 Moving towards a sustainable carbon footprint

So far, you've been considering reductions in average individual or household carbon footprints by 20% to 30% or more.

But it is becoming increasingly clear that this will not be enough. As I mentioned in Section 4, developed countries, like Britain, Germany and America, will have to reduce their CO2e emissions by 60% to 80% or more by 2050 to prevent climate change running out of control, while at the same time allowing the growing populations of Africa, India and China to r
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4.3 Technical and behavioural actions

The numbers generated by the carbon calculator use a computer model based on some of the best information available. However, as I mentioned earlier, the results are not exact because calculators typically require you to enter broad categories of information about yourself and your household. And there are always uncertainties about some of the data on which the calculator is based. Nevertheless, the calculator allows you to explore the important actions needed to lighten your carbon load and
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2.2 Records of the Earth's temperature

To put the temperature records reported by the IPCC in context, we start with a longer-term geological perspective on the Earth's GMST.

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1.5 ‘Radiative forcing’ as an agent of climate change

Since its first major report in 1990, the IPCC has used the concept of â€˜radiative forcingâ€™ as a simple measure of the importance of a potential climate change mechanism. The basic idea is straightforward. Any factor that disturbs the radiation balance at the top of the atmosphere has the potential to â€˜forceâ€™ the global climate to change: it will either warm up or cool down until a balance is restored. The perturbation to the energy balance of the whole Earth-atmosphere system i
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1.4.3 It's all down to connections

For Iris Marion Young, the responsibility of those in North America and Europe towards distant others does indeed rest with their connections to injustices elsewhere, but it would be a mistake to stretch this line of reasoning too far. Although these connections, whether as a consumer, boardroom executive or shop manager, can establish a line of responsibility, as was claimed in Section 3.1, for Young this is only the starting point and not the end point of our involvement. We do not have to
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1.3.6 Activity 4

## Activity 4

Nike drew up its code of conduct, as I have indicated, to meet its own concerns. Cast your eye down the checklist in Extract 2 and give yourself time to consider what issues might have been added if the code had been drawn u
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2.3.1 Public participation and perspectives on sustainability

When it comes to issues around the environment, â€˜expertsâ€™ â€“ whether ecologists, economists or other types of social scientist â€“ are clearly not infallible. Environmental crises have led to a questioning of traditional expert support as a guarantor of environmental planning. This has had two consequences. Firstly, there have been some interesting and useful explorations amongst environmentalists in seeking guarantors through the domain of spiritualism. In particular, traditional worldv
Author(s): The Open University

References

Allison, L. (1991) Ecology and Utility: The Philosophical Dilemmas of Planetary Management, Leicester, Leicester University Press.
BBC (2008) â€˜The wrong way to a warmer world?â€™ [online], BBC News, 3 April, http://news.bbc.co.uk/1/hi/programmes/analysis/7328634.stm (accessed 15/4/10).
van den Born, R.J.G. (2008) â€˜Rethinking nature: public visions in the Netherlan
Author(s): The Open University

## Author

This unit was prepared by Tom Power with guidance from Dr Arlene Hunter.

Tom Power is a lecturer in science education at The Open University. His research interests include teacher education in the global south (www.open.ac.uk/deep) and the CASE intervention. He has been a teacher and an advisory teacher in East Sussex and a specialist adviser to the TTA teacher research panel.

Dr ArlÃ«ne Hunter, Staff Tutor in Science in Ireland
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Analyses of over 400 proxy climate series (from trees, corals, ice cores and historical records) show that the 1990s was the warmest decade of the millennium and the 20th century the warmest century. The warmest year of the millennium was 1998, and the coldest was probably 1601. (Climatic Research Unit, 2003)

Throughout historical times, fluctuations in the Earth's mean temperature have been recorded. During the seventeenth century, the Thames periodically froze over during winter and m
Author(s): The Open University