The concept of field

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.

When is it useful to define a field?

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:

Scalar fields

Vectors are pretty complicated since they tell both about magnitude and direction. Let's consider an example that only specifies a single number (though one that can be positive or negative): the 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 a function that gives the temperature at a longitude $x$ and latitude $y$, $T(x,y)$, 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 an airplane the temperature drops. It is also 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.

We also know well 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 weather vane (shown on the left) and an anemometer (shown on the right) and measure the direction and speed of the wind. 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.

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). 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 map (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"

We use mathematics in science to express the idea of relationships.

Often, two measurable quantities in science are 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 just a rule for taking one quantity and generating another.

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)

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 that region of space . 

This can be particularly confusing if the field is zero 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. (Besides, the dimensions are wrong. An area has dimensionality of L². A field can have any dimensions.) 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.

Joe Redish 2/11/12
Wolfgang Losert 2/16/13