Scalar and vector fields

Imagine what the temperature might be in the different parts of the room you're sitting in now.

Some parts of it, maybe near the door or windows, will probably be cooler, while other parts, maybe near a heater, will be warmer. And in between these regions of course there must be a continuous smooth change in temperature.

This quantity "temperature", let's call it T, therefore has various different values throughout that three-dimensional space that you're sitting in. Let's describe position by the three Cartesian coordinates x, y and z.

So at any given position (x,y,z) the temperature T has a particular value, and if we change that position then T will probably change too. In other words T is a function of x, y and z and we can write T(x,y,z).

This means that T is a scalar field.

Now imagine the air moving around in that room you're in. In some parts it will be moving quickly, above the heater maybe, or near an open window, or near your nose, while in other parts it will be moving slowly.

The quantity describing that air movement is "velocity", let's call it v. That quantity v also has a different value at different positions, so we can write v(x,y,z) and this quantity too is a field.

However there is one important difference between these two fields temperature and velocity. The velocity field has a direction at each point, whereas the temperature does not.

At any position (x,y,z), the air at that point is moving in a particular direction, with a particular speed, so there are two separate pieces of information.

The temperature at that position just has a value, 20 degrees say, there is only one piece of information. There is no direction associated with that temperature.

(There is a direction in which the temperature is increasing from that point, but that is the temperature gradient field, which is a different quantity and which is indeed a vector field.)

We therefore say that the velocity is a vector field, and the temperature is a scalar field.