Introduction to the calculus of variations
This free course concerns the calculus of variations. Section 1 introduces some key ingredients by solving a seemingly simple problem – finding the shortest distance between two points in a plane. The section also introduces the notions of a functional and of a stationary path. Section 2 describes basic problems that can be formulated in terms of functionals. Section 3 looks at partial and total derivatives. Section 4 contains a derivation of the Euler-Lagrange equation. In Section 5 the Euler
Author(s): Creator not set

Plans of houses and instructions for assembling shelves, etc., often come in the form of scale diagrams. Each length on the diagram represents a length relating to the real house, the real shelves, etc. Often a scale is given on the diagram so that you can see which length on the diagram represents a standard length, such as a metre, on the real object. This length always represents the same standard length, wherever it is on the diagram and in whatever direction.

Author(s): The Open University

The data in Table 5, which are given for several countries in Europe and elsewhere, are the average annual alcohol consumption in litres per person and the death rate per 100 000 of the population from cirrhosis and alcoholism. It would seem obvious that the two are related to each o
Author(s): The Open University

Self-assessment questions are intended to test your immediate comprehension of a reading section. If you have difficulty with them you should read again the appropriate parts of the material. Before checking the solution to any part of a question, you should work through all the parts of that question.

## Activity 24

A new train operator boasts ‘Train times reduced by 12%’. Decrease 90 minutes by 12%. Give your answer as minutes and seconds.

After studying this course, you should be able to:

• work with simple ratios

• convert between fractions, decimals and percentages

• explain the meaning of ratio, proportion and percentage

• find percentages of different quantities

• calculate percentage increases and decreases.

Author(s): The Open University

The so-called ‘content/process’ debate in mathematics involves discussion of the relative importance of content and process in mathematics. It originated as part of a discussion about the nature of mathematics, particularly of school mathematics, and of the purposes for which mathematics is learned. Identifying content and process in mathematics draws attention to the idea that mathematics is a human activity.

As a teacher of mathematics in the UK, you are faced with a national curr
Author(s): The Open University

Microbes – friend or foe?
Microbes often get a bad name. Whilst some of them do cause disease, others play vital roles in recycling nutrients in the soil to enable plants to grow, and in breaking down human waste. Without microbes, we would have no beer, no yoghurt, no coffee. That's quite impressive for something too small to see. This free course, Microbes friend or foe? sheds some light on them.
Author(s): Creator not set

Environment: treading lightly on the Earth
Environment: treading lightly on the Earth focuses on the problem of greenhouse gas emissions, especially carbon dioxide. This free course will give you an understanding of the nature and importance of carbon footprints of individuals and households. It will enable you to measure your own carbon footprint and explore what you could do to reduce that footprint and so ‘tread more lightly on the Earth’.Author(s): Creator not set

Broham, J. (1996) ‘Postwar development in the Asian NICs: does the neoliberal model fit reality?’, Economic Geography, vol.72, pp. 107–30.
Castree, N., Coe, N.M., Ward, K. and Samers, M. (2004) Spaces of Work: Global Capitalism and Geographies of Labour, London, Sage.

Holding up the East Asian success story as the way forward has, as I indicated above, little appeal for the antisweatshop movement. For its members, a different image comes to mind of thousands of workers eking out a living from the numerous sweatshops which dot that part of the world: one that involves the perpetuation of poverty wage levels, the use and abuse of poor communities, and the constant taking advantage of what is ready to hand, followed by withdrawal and abandonment. What they se
Author(s): The Open University

Significantly, no one from the pro-market lobby is actually denying that sweatshops exist, or trying to cover up the fact that workers in such places have to endure bad working conditions. But, as the subtitle of Krugman's (1997) article suggests: ‘bad jobs at bad wages are better than no jobs at all’. Low as the wages are in the offshore T-shirt or microwave factories compared with those in more developed economies, they tend to be higher than those of other workers around them. The huma
Author(s): The Open University

A particularly important set of debates arises in relation to different notions of distributive justice. Do notions of distributive justice apply to the rights of individuals and the acts that they commit, or do they also apply to states of affairs, to the pattern of the results arising from those actions? In the former case, an outcome is just or unjust if it arises from just or unjust actions; whereas in the latter, the principles of justice apply to the pattern of outcomes. This latter not
Author(s): The Open University

Apart from the internal redistribution of income resulting from greater exposure to the world economy, the effects of one of the UR agreements in particular have achieved a certain notoriety because the agreement clearly imposes huge costs on farmers and consumers in developing countries, to the benefit of corporations in developed countries. This is the agreement on Trade-Related Aspects of Intellectual Property Rights (TRIPS), which strengthens international rules governing patents, tradema
Author(s): The Open University

Cyborgs and cybernetics
What are cyborgs? Would a cyborg future deliver positive human advances or a Hollywood-style nightmare in which human beings have become a sub-species? Could we one day download our minds? This album gives an insight into the development of cybernetics and how it is used to fuse technology and humanity. The interfaces that communicate between man and machine are developing rapidly and to Prof. Kevin Warwick at Reading University, cyborgs are a technological evolutionary step forward from humans.
Author(s): The OpenLearn team

Structural Integrity: Materials Testing
How is safety built into the design of new structures? What sort of tests are used to ascertain the safety of proposed designs? Structural integrity, the study of the safe design and assessment of materials and structures under load, has become crucial in engineering design. Concepts within stress analysis have wide applicability, as there are very few manufactured components and products that do not experience any loading during their life. The tracks on this album demonstrate a selection of
Author(s): The OpenLearn team

Water Treatment
Do you think about where your water comes from? In the UK each of us uses an average of about 150 litres of water per day! The seven video tracks in this album consider issues of demand and quality in water supply as well as treatment processes. They give information on methods of minimising waste, emergency water treatment and effluent control. This material forms part of T308 Environmental monitoring, modelling and control.Author(s): The OpenLearn team

Innovation Design: Sustainable Communities
Can you picture the future in a world without fossil fuels? Perhaps you think that living an "alternative" lifestyle has to mean painful and radical changes to the way you live now. This album looks at various small scale initiatives which show that living sustainably may not be as unpalatable as you might imagine. Bedzed, Findhorn, Hockerton, Samsoe Island; these are all decentralised communities at the forefront of a major social and technological experiment. In our world of finite resources a
Author(s): The OpenLearn team

Digital Film School
Have you ever wanted to pick up a video camera and make a short video or film, but felt intimidated by your lack of knowledge? The explosion of film-making for websites and mobiles gives people and organisations the opportunity to tell their stories and show what they have to offer, at low cost. This collection of exciting videos features The Open University’s experienced team of filmmakers, who will show you some of the craft secrets that underpin good filmmaking, and how professionals stay
Author(s): The OpenLearn team

In this course we examine one factor that very often seems to be found skulking around close to problems and solutions: temperature.

Almost whatever we do, wherever we are, temperature changes. Stay in the same spot and you'll find daytime and night-time temperatures can be markedly different. You may even find significant changes in temperature during the day. When moving you can encounter more rapid variations. For example, an aircraft might leave a tropical runway where the air tempe
Author(s): The Open University