The content acknowledged below is Proprietary (see terms and conditions) and is used under licence.

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

After studying this unit you should:

know some basic definitions and terminology associated with scalars and vectors and how to represent vectors in two dimensions;

understand how vectors can be represented in three (or more) dimensions and know both plane polar and Cartesian representations;

know ways to operate on and combine vectors.

This unit introduces the topic of vectors. The subject is developed without assuming you have come across it before, but the unit assumes that you have previously had a basic grounding in algebra and trigonometry, and how to use Cartesian coordinates for specifying a point in a plane.

This is an adapted extract from the Open University course *Mathematical methods and models* (MST209)

After studying this unit you should:

be able to solve homogeneous second-order equations;

know a general method for constructing solutions to inhomogeneous linear constant-coefficient second-order equations;

know about initial and boundary conditions to obtain particular values of constants in the general solution of second-order differential equations.

The content acknowledged below is Proprietary (see terms and conditions) and is used under licence.

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

After studying this unit you should be able to:

understand how the wave and diffusion partial differential equations can be used to model certain systems;

determine appropriate simple boundary and initial conditions for such models;

find families of solutions for the wave equation, damped wave equation, diffusion equation and similar homogeneous linear second-order partial differential equations, subject to simple boundary conditions, using the meth

During this unit you will:

learn some basic definitions and terminology associated with differential equations and their solutions;

be able to visualize the direction field associated with a first-order differential equation and be able to use a numerical method of solution known as

*Euler's method*;be able to use analytical methods of solution by direct integration; separation of variables; and the integrating factor method.

First-order differential equations

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 *

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

*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.

*<*

Author(s):

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Copyright 2009 University of Nottingham

In the following subsections, we apply the vector ideas
introduced so far to *displacements* and *velocities*. The
examples will feature directions referred to points of the compass,
known as *bearings*.

The direction of Leeds relative to Bristol can be described
as ‘15° to the East of due North’,
or N 15° E. This is an instance of a
**bearing**. Directions on the ground are typically given like
this, in terms of the directions North (N),
South (S), East (E)

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.

On completion of this unit you should be able to:

convert a vector from geometric form (in terms of magnitude and direction) to component form;

convert a vector from component form to geometric form;

understand the use of bearings to describe direction;

understand the difference between velocity and speed;

find resultant displacements and velocities in geometric form, via the use of components.

In this unit you will see first how to convert vectors from geometric form, in terms of a magnitude and direction, to component form, and then how conversion in the opposite sense is accomplished. The ability to convert between these different forms of a vector is useful in certain problems involving displacement and velocity, as shown in Section 2, in which you will also work with bearings.

This unit is an adapted extract from the Open University course

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

1.3.2 Search engines and subject gateways

Although both search engines and subject gateways will help you find the resources that you need, the types of information that you find will differ.

Search engines such as Google and Yahoo! search the internet for keywords or phrases, and then show you the results. These results are not mediated by the search engines, and therefore you need to use your own judgement on the reliability of the results. You may, for example, find websites written by experts, alongside websites written by

4.3.2 Complementary currencies

**Complementary currencies** also demand a rethink of our economy, but have a more imaginative and radical edge. Because of the difficulties with conventional monetary systems, various alternatives are being tried. These are usually restricted to a particular group of people, and so are called ‘local’ or ‘complementary’ currencies. They are generally based in a local community and enable people to exchange goods and services without resorting to ‘traditional’ currency. Some are

Communities such as Findhorn already behave as if natural resources need careful management: they work hard to reduce fossil fuel use. A central assumption of this way of thinking is that people need to root economies more locally (Figure 15). To see the same impulse spread through the mainstream economy would require that th

2.2 Vibrant civil societies and a networked globe

One thing is common to all three attempts to find a route to a sustainable economy and society: in different ways they all assume that people will get actively involved in making human societies more sustainable. But this transformation will not take place through the corporate world's promises, by local protectionism, a return to ‘strong states’ or the publication of numerous indicators. Any of the three positions outlined above requires interactions and feedbacks created by a vibrant

2.1 Political responses to climate change and the environment

Not for the first time in this book, you are faced with a term that is important but difficult to define precisely. Although the fact that plenty of people from different standpoints are using the term ‘globalisation’ is some measure of its importance, it can be confusing to find that there are different ways of framing what it means for humans and the environment today and in the future. In this section, the range of political responses to climate change and environment–economy interac