This chapter provides examples of how to apply the tensor concepts contained in Chapters 4 and 5, just as Chapter 3 provided examples of how to apply the vector concepts presented in Chapters 1 and 2. As in Chapter 3, the intent for this chapter is to include more detail about a small number of selected applications than can be included in the chapters in which tensor concepts are first presented.

The examples in this chapter come from the fields of Mechanics, Electromagnetics, and General Relativity. Of course, there's no way to comprehensively cover any significant portion of those fields in one chapter; these examples were chosen only to serve as representatives of the types of tensor application you're likely to encounter in those fields.

The inertia tensor

A very useful way to think of mass is this: mass is the characteristic of matter that resists acceleration. This means that it takes a force to change the velocity of any object with mass. You may find it helpful to think of moment of inertia as the rotational analog of mass. That is, moment of inertia is the characteristic of matter that resists angular acceleration, so it takes a torque to change the angular velocity of an object.

Many students find that rotational motion is easier to understand by keeping the relationships between translational and rotational quantities in mind.

The vector concepts and techniques described in the previous chapters are important for two reasons: they allow you to solve a wide range of problems in physics and engineering, and they provide a foundation on which you can build an understanding of tensors (the “facts of the universe”). To achieve that understanding, you'll have to move beyond the simple definition of vectors as objects with magnitude and direction. Instead, you'll have to think of vectors as objects with components that transform between coordinate systems in specific and predictable ways. It's also important for you to realize that vectors can have more than one kind of component, and that those different types of component are defined by their behavior under coordinate transformations.

So this chapter is largely about the different types of vector component, and those components will be a lot easier to understand if you have a solid foundation in the mathematics of coordinate-system transformation.

Coordinate-system transformations

In taking the step from vectors to tensors, a good place to begin is to consider this question: “What happens to a vector when you change the coordinate system in which you're representing that vector?” The short answer is that nothing at all happens to the vector itself, but the vector's components may be different in the new coordinate system. The purpose of this section is to help you understand how those components change.

The real value of understanding vectors and how to manipulate them becomes clear when you realize that your knowledge allows you to solve a variety of problems that would be much more difficult without vectors. In this chapter, you'll find detailed explanations of four such problems: a mass sliding down an inclined plane, an object moving along a curved path, a charged particle in an electric field, and a charged particle in a magnetic field. To solve these problems, you'll need many of the vector concepts and operations described in Chapters 1 and 2.

Mass on an inclined plane

Consider the delivery woman pushing a heavy box up the ramp to her delivery truck, as illustrated in Figure 3.1. In this situation, there are a number of forces acting on the box, so if you want to determine how the box will move, you need to know how to work with vectors. Specifically, to solve problems such as this, you can use vector addition to find the total force acting on the box, and then you can use Newton's Second Law to relate that total force to the acceleration of the box.

To understand how this works, imagine that the delivery woman slips off the side of the ramp, leaving the box free to slide down the ramp under the influence of gravity.

Introduction

Every commercial kitchen must operate using a sound basis for costing and control of raw materials such as food, cleaning, wages and overheads. The principles are the same whether the food service is run at a hospital, aged-care home, institution, take-away shop or restaurant. The capacity of a restaurant to make a profit, make a fair financial return on the investment and reward the effort involved is very much dependent upon effective control of the day-to-day running costs. Likewise, hospitals, aged-care facilities and other food services need to operate within strict financial budgets.

Effective cost control has been described as the identification and regulation of operating costs. These will be influenced by the style and systems of service to be offered. Food and labour costs make up the largest proportion of operating costs. In this chapter we examine food cost.

An acceptable budget for food is calculated by costing menus and recipes. This involves choosing suppliers carefully after reviewing all the options. Suppliers need to know the purchasing standards that are required for the food. These will describe quality, size, colour, fat content, age and origin (organic or halal, for example). Is the supplier able to offer continuity of supply at an acceptable price? Does the supplier stock a range of food items needed for the menu? Buying from a large number of suppliers makes it difficult to build relationships, takes time attending to deliveries and increases accounting costs.

Introduction

A well-organised kitchen starts with the preparation of basic ingredients. Every chef knows that it is the quality of the first two hours’ work that sets the standard for the food service that follows. Things left undone or food poorly prepared during this preparation time can give rise to declining standards of cooking, poor presentation of dishes, frayed tempers at the busiest time of the service and sometimes lengthy delays between courses.

Getting things ready is referred to as mise en place (‘putting in place’). You can tell a well-run, happy kitchen by the way it organises its mise en place. It does not matter whether it is located in a restaurant, hospital or cafeteria – just before the time for service, the bustle of the kitchen will die down for 5 or 10 minutes. During this quiet lull you will see the work benches clean and tidy, dishes of food arranged, seasoning boxes filled and saucepans gently steaming or the food in the bain-marie neatly laid out, ready for plating. The hot cupboards will be hot and neatly stacked with plates and special dishes. Cold plates and dishes will also be ready for the serving of cold food.

This chapter describes the work involved in this preparation. The menu and type of service will determine what ingredients are selected and how they are prepared. In a well-run kitchen, preparation for the next meal will be started as soon as the current meal service is concluded.

Introduction

Vegetables and fruit provide an excellent source of vitamins, minerals and dietary fibre. The enormous variety of produce available today offers an opportunity for creativity in preparing dishes for the menu. Vegetables may feature on a menu on their own, as an accompaniment to main dishes or mixed in with other ingredients. Fruits are increasingly used throughout the menu and not just reserved for desserts.

For the restaurant industry, unusual vegetables or varieties grown specifically to a certain size or shape can provide exclusivity. The decision to purchase fresh produce under-ripe, ripe or very ripe will depend on how and when the items are to be used. Large-size vegetables may be easier to cut into shapes but may not be as flavoursome as smaller sizes.

The increase in variety of vegetables and fruit available in Australia is largely due to the fact that established growing areas have continued to be productive and new areas have been developed in the country's northern regions. Growers can choose to crop a variety of produce suited to cool, temperate or tropical conditions. We can usually draw supplies from these areas at all times of the year. New varieties of vegetables and fruit are being grown in commercial quantities, and when local supply is unavailable produce is imported from overseas.

The right time to purchase vegetables and fruit is when they are in season: at their peak in quality, plentiful in supply and cheapest. The drawback of purchasing out-of-season produce is a lack of flavour and higher cost.

Introduction

Until recently, the seasons dictated the availability of food. During long winters, food was scarce, so the abundance of food from the summer and autumn harvests was preserved to be used in the winter months. In order to remain edible, food had to be protected from damage from a variety of sources: vermin, insects, unfavourable temperatures, exposure to high levels of moisture, oxygen in the air, yeasts and moulds, harmful enzymes and micro-organisms (bacteria). Traditional methods for controlling microbial spoiling included:

• limiting water availability by dehydration or the addition of sugar or salt

• changing the pH by pickling

• destroying micro-organisms using heat; by smoking or adding alcohol

• using micro-organisms to lengthen the shelf life of milk by making cheese and yoghurt

• fermenting grains and fruits to make wine, beer, cider and bread.

Modern technologies have added to that list of methods. There are now more precise heat processes, like pasteurisation, and better ways of removing water, by freeze-drying and spray drying. There is easy access to cold preservation using refrigeration and freezing, and vacuum and modified-atmosphere packaging. Antioxidants inhibit the onset of rancidity, and advances in the use of food additives and irradiation are available to destroy micro-organisms.

Thanks to these advances in food technology, seasonal foods are available at any time of the year in the required quantities and quality across the nation.

Introduction

Traditionally, ‘game’ was a term used to describe food obtained from animals that were either hunted or trapped. As these were wild animals, their diets varied from season to season and this contributed to their unique flavour. Today, however, much of what was originally termed game is farmed to ensure supply for the restaurant trade and as such, the animals’ diets are controlled and much of their unique flavour has been lost.

Game is broadly described as being feathered or furred. Feathered game includes pheasant, quail, partridge, guinea fowl, emu, ostrich, pigeon (squab) and wild duck. Furred game includes venison (deer), rabbit, hare, kangaroo, wild boar and buffalo. Other types of game available in Australia are crocodile, camel, witjuti grubs, brush tail possum, Yolla (also known as muttonbird), snails and frogs legs.

All game meats for human consumption should be purchased from accredited suppliers, to ensure compliance with the relevant processing and regulatory guidelines. Game meats are obtainable as primary and secondary cuts, and no dressing (plucking, skinning or drawing) of the meat is required. Cuts are available fresh, chilled or vacuum packed in various combinations to suit menu dishes. Venison can be ordered ‘de-nerved’; that is, with the silvery connective tissues that covers the muscles removed.

Game is highly perishable; furred game should be stored in the same way as butcher's meat and feathered game in the same way as poultry. See Chapters 18 and 19 for the storage requirements.