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.

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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 a = a 1 i + a 2 j, what are its magnitude |a| and direction θ?

The first part of this question is dealt with using Pythagoras’ 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

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 Eisbär (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 – eminently sensible. The Latin name for th
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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
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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
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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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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 ‘radiative forcing’ 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 ‘force’ 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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3 Recorded temperatures

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
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2 A 4.6 billion-year history

Climate change is a natural process of warming and cooling that has occurred all through the Earth's history. Throughout geological time there have been ‘hot-house’ periods and ice ages. In order to understand the current situation, it is necessary to have some sense of context and perspective, from historical and geological time-scales. The document below shows a chart showing a generalised temperature history of the Earth.

Click on 'View document' to see the chart

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1 Natural climate change?

The chart below shows a record of the global mean surface temperature of the Earth compiled for the past 140 years. Clearly there is an upward trend, but what does a chart like this really show?

References

Anon. (2003) ‘Spy chief warns food industry over terrorism’, Environmental Health News, 24 October 2003, p. 2.
Cabinet Office (2003) Dealing with Disaster, revised 3rd edn, Civil Contingencies Secretariat.
Commercial Union Risk Management Ltd (1992) ‘Crisis: A timetable for recovery’.
Dodswell, B. (2000)
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6 Conclusion

Perhaps it is a truism to say that all life is full of risk. We encounter many uncalculated outcomes, some beneficial and others adverse. It can be difficult to know which adverse events will prove permanently disadvantageous, since some may lead to innovation and opportunities for the future. Businesses, especially in the financial context, often consider risk in terms of opportunities for gain. Risk in our context is a way of describing the probability and consequences of harm, or at worst
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5.3.4 Plan testing and validation

It is one thing to have a plan; it is another thing to have a plan that you can rely on to work. There is an old military maxim that ‘A plan only gets you into first contact with the enemy. After that, you fly by the seat of your pants’ (Anon). A 1993 IBM report on business continuity planning confirmed this when it revealed that ‘half of the plans failed completely or substantially when they were first tested’ (IBM, 1993, p. 5).

The IBM report identified three categories of pla
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5.3.2 Plan auditing

Having got the draft plan, it is worth checking it over to see that all the major issues have been covered. The appendix below contains a set of guidelines for the initial audit of a generic ‘general purpose’ plan. For site-specific plans such as might be produced by an SHE manager in industry, or a business continuity manager for an office complex, the headings may need some modification.

Guidelines for an emergency response plan audit (PDF, 2 pages, 0.1MB)

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5.7.1 Plan preparation

Perhaps the first question to ask is ‘What is an emergency plan?’ Dodswell, in his guide to business continuity management, defined an ‘emergency management plan’ as simply:

A plan which supports the emergency management team by providing them with information and guidelines.

(Dodswell, 2000, p. 56)

Another definition, of an ‘emergency preparedness plan’ prepared in the co
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