5.1 Identifying edges of a polygon In this section, we revisit the construction of surfaces by identifying edges of polygons, as described in Section 2. Recall that, if we take any polygon in the plane and identify some of its edges in pairs, then we obtain a surface. When specifying how a given pair of edges is to be identified, we choose one of the two possible re
2.1 Surfaces in space In Section 2 we start by introducing surfaces informally, considering several familiar examples such as the sphere, cube and Möbius band. We also illustrate how surfaces can be constructed from a polygon by identifying edges. A more formal approach to surfaces is presented at the end of the section. Figure 3 shows
Modelling heat transfer The main teaching text of this course is provided in the workbook below. The answers to the exercises that you'll find throughout the workbook are given in the answer book. You can access it by clicking on the link under the workbook. Click the link below to open the workbook (PDF, 0.4 MB). Click the link below to open the answerbook (PDF, 0.2 MB).<
1 Analysing skid marks The main teaching text of this course is provided in the workbook below. The answers to the exercises that you'll find throughout the workbook are given in the answer book. You can access it by clicking on the link under the workbook. Click the link below to open the workbook (PDF, 0.2 MB). Click the link below to open the answerbook (PDF, 0.1 MB).<
Learning outcomes After studying this course, you should be able to: create simple models, given a clear statement of the problem identify the simplifying assumptions that underpin a model identify the key variables and the parameters of a model apply the input–output principle to obtain a mathematical model, where appropriate.
Try some yourself 1 Calculate the area of a carpet in a model house if the real carpet has an area of 22 m2. On the scale used, 1 cm represents 0.25 m. (a) Since 1 3.3 Volumes What is a volume? The word usually refers to the amount of three-dimensional space that an object occupies. It is commonly measured in cubic centimetres (cm3) or cubic metres (m3). A closely related idea is capacity; this is used to specify the volume of liquid or gas that a container can actually hold. You might refer to the volume of a brick and the capacity of a jug – but not vice versa. Note that a container with a particular volume will not nec Try some yourself 1 Find the area of a circle of (a) radius 8 cm, and (b) radius 15 m. (a) Try some yourself 1 Find the area of each of these shapes. 2.4 Drawing circles Drawing circles freehand often produces very uncircle-like shapes! If you need a reasonable circle, you could draw round a circular object, but if you need to draw an accurate circle with a particular radius, you will need a pair of compasses and a ruler. Using the ruler, set the distance between the point of the compasses and the tip of the pencil at the desired radius; place the point on the paper at the position where you want the centre of the circle to be and carefully rotate the compass Try some yourself Find the area of a circle of (a) radius 8 cm, and (b) radius 15 m. (a) 3 Aims The aim of this section is to help you to think about how you study mathematics and consider ways in which you can make your study more effective. 3 Group axioms Section 3 is an audio section. We begin by defining the terms group, Abelian group and order of a group. We then demonstrate how to check the group axioms, and we extend the examples of groups that we use to include groups of numbers – the modular arithmetics, the integers and the real numbers. Click the link below to open Section 3 (11 pages, 703KB). Introduction In this unit we use the geometric concept of symmetry to introduce some of the basic ideas of group theory, including group tables, and the four properties, or axioms, that define a group. Please note that this unit is presented through a series of PDF documents. This unit is an adapted extract from the Open Unviersity course Pure ma 3 The language of proof In Section 3 we examine the language used to express mathematical statements and proofs, and discuss various techniques for proving that a mathematical statement is true. These techniques include direct proof, proof by mathematical induction, proof by contradiction and proof by contraposition. We also illustrate the use of counter-examples to show that a statement is false. Click the link below to open Section 3 (17 pages, 374KB). Introduction When we try to use ordinary language to explore mathematics, the words involved may not have a precise meaning, or may have more than one meaning. Many words have meanings that evolve as people adapt their understanding of them to accord with new experiences and new ideas. At any given time, one person's interpretation of language may differ from another person's interpretation, and this can lead to misunderstandings and confusion. In mathematics we try to avoid these difficulties by ex 4.2 Defining useful subsets of the complex number system, and proving the Nested Rectangles Theorem You will no doubt recall that in real analysis extensive use is made of the modulus function 7 Solutions to the exercises Section 6 contains solutions to the exercises that appear throughout sections 1-5. Click the link below to open the solutions (13 pages, 232KB). Learning outcomes After studying this course, you should be able to: understand the definition of a real function use the notation for intervals of the real line recognise and use the graphs of the basic functions described in the audio section understand the effect on a graph of translations, scalings, rotations and reflections understand how the shape of a graph of a function features properties of the function such as increasing, decr 3.1.1 Try some yourself 1 Express each of the following percentages as fractions: (a) 40% (b) 8% (c) 70% (d)
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. It gives us a way of measuring the “closeness” of two numbers, which we exploit in writing expressio
