Class Content I >Modeling with mathematics > Math recap > Functions and Functional Dependence
2.2.4.3
Prerequisites:
You've probably heard of "electric and magnetic fields" and maybe even about "gravitational fields". What are fields and why do we need them? Basically, the field is just a particular kind of function. To get a better sense of it, we will introduce a few examples of fields you already know, and then get into the mathematics of fields.
Let's start with an example where having a field is useful: Say we want to calculate the forces and motion of a charged object in the presence of multiple other stationary charges. As our object of interest moves, the forces it experiences from all other (stationary) charges changes. But at each position, the force our object experiences is well defined  we simply have to add the electric forces that all other objects in the system exert on our test object. So essentially we could calculate a force that our object would experience for any point in space. The concept of fields is a very useful mathematical concept that helps us represent such quantities that are defined for all points in space (e.g. in our example "force due to stationary charges"). In fact you are very familiar with fields in everyday life:
As an example, consider a temperature field. A temperature field is an assignment of a temperature to every point in space. If we are considering the temperature near ground level, we might represent this function, T(x,y), by colors as shown in the weather map at the right; a temperature map of the US on a cool February morning.


Of course, the temperature is also a function of height. As you go up in a plane the temperature drops  and it is important to know the temperature as a function of height as well as of latitude and longitude if you are going to try to analyze weather patterns. If you are trying to decide what to wear when you go out, the map above might be enough.
Of course we know well enough that the temperature depends not just on position in space but on time. Often our fields will depend on both space and time. We would write T(x,y,z,t) if we wanted to emphasize all the things our temperature field depends on.
Vector fields: Multiple representations
The temperature field discussed above is a scalar field  that is, it's just a single number (though it can be positive or negative). Many of the fields that we will be using in this class are vector fields  that is, we are assigning not just a number to each point in our spatial domain, but a vector. A nice example of this continuing our weather context is the wind. At each point in space you might take a little anemometer (wind gauge) and measure the direction and speed of the wind. (To get the direction, you would change the orientation of the anemometer until you found the direction where the speed the cups were driven was the greatest.) At each point in space we would attach a vector giving the speed and direction of the wind at that point. There are a lot of different ways to represent this. Three are shown in the diagrams below.

Public domain photo

In the first map (of the northeastern US) the magnitude of the wind speed is given by the color and the direction by a label (using the letters N, E, S, W from the compass rose). In the second map (of the Arabian peninsula), the direction is given by a local arrow and the speed by a number. In the third (of the British isles) the direction is given by the direction of an arrow and the speed is represented by the arrow's thickness. Other representations are used as well.
We will be using vector fields in this class to describe how the forces felt by an object varies as the object changes its position  particularly gravitational and electrical fields.
The mathematics of fields
The mathematical idea of "function"
One of the most important things we use mathematics for in science is to express the idea of relationships. Often, two measurable quantities in science will be related in some way. One may "cause" the other; we may be able to change the value of the first and as a result change the value of the second. A stronger push may yield a larger acceleration. Or, two measurable quantities may combine to yield a third observable. The period of a mass hanging on a spring depends both on the object's mass and on the "stretchability" (spring constant) of the spring. In representing these ideas with a mathematical model, the key idea is that of function. In math, a function is a rule for taking one quantity and generating another. This is represented by the diagram below.
The set of objects on which the function is defined is called the domain and the set of values it can give back is called the range. The mapping rule is called the function. In this case we are mapping from the set of Latin characters to the set of Greek characters. So if we called our function, F, then we might write F(A) = α, F(B) = β, and so on. Notice that not every element in the set of Latin characters has to belong to the domain of our function. There is no Greek analog to the letter "J". And not every element in the set of Greek characters has to be the image of something coming from Latin. The character "Ψ" has no Latin analog. The domain and range are not necessarily the whole "natural" set. They are properties of the function, where it is defined and what values we give.
Note that in general, a "function" does not have to start with a number and end with a number. We can have a function of anything we do math with that gives back anything we do math with: any set of objects, an integer, a real number, a complex number, a set, a set of numbers, even a function!
Field: A function of position (and maybe time)
In the specific case when our domain is space  that is, when we want to describe how some quantity varies throughout space  then the specific function is called a field. This choice of term is somewhat unfortunate. "Field" is a technical term, so it does NOT have the everyday meaning that you might naturally infer. In everyday speech, "field" refers to a restriction of a space  a particular bit of space; or a particular choice of career; a restriction of a different kind of "space". But our mathematical term DOES NOT REFER TO A RESTRICTED REGION OF SPACE. Rather, it refers to THE FUNCTION THAT IS DEFINED ON A RANGE OF SPACE.


This can be particularly confusing if the field is 0 or negligibly small except in some region of space. Then it might seem natural to identify the "field" with the region of space. Don't do this. It causes serious confusion.
It's like giving the volume of a room when what was wanted was the pattern of temperatures in that room. Doesn't help at all if you are trying to find out where it's too hot and too cold.
Followons:
Joe Redish 2/11/12
Wolfgang Losert 2/16/13
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