What are vector fields?

While looking at the leak in Charlie’s kitchen, Larry drops some ink from a pen into the puddle and talks a bit about “vector fields”. What are they?

Well, for some things we want to measure, we need two kinds of information: their size or magnitude and their direction. One example is a force. The magnitude of the force is how strong it is; maybe this would be measured in pounds (or newtons in the metric system). For example, someone may shove you, or a wind might push you, with a certain number of pounds of force. But for a force, you have to know more than the magnitude: you have to know the direction. So, a force may push you sideways, or forward, or upward, or some direction in between. So a force is described by two things: its magnitude and its direction.

In physics we talk about another quantity like this: velocity. To know a velocity you need to know its magnitude; for example, 130 meters per second. However, you usually need to know what direction you’re travelling in. So a velocity might be described as: 130 meters per second in the direction 30 degrees north of east.

Quantities that need a magnitude and direction to describe them completely are called vectors. There is a nice way to picture vectors: as arrows. The length of the arrow gives its magnitude (when we know the scale) and the direction the arrow points gives the … well…direction. Thus, the velocity described above might be represented as an arrow 130 centimeters long, pointing 30 degrees north of the due east direction.

So what is a vector field? A vector field is a bunch of vectors drawn at a lot of different points. They represent forces or velocities at those points. Here’s an example.

Suppose you have a fish tank and start pumping water into it. (To make sure that it doesn’t overflow and make a mess, you may want to drain water out at the same rate). Now as the water flows in, it swirls around, creating currents in the tank. At different places in the tank the water is moving in different directions and at different speeds. If you had a lot of highly trained fish, you could get them to swim to different positions in the tank and measure the speed and direction of the water at some instant at all these different positions. Each fish could then hold an arrow at its position, with the length of the arrow representing the magnitude of the velocity of the water there, and with the arrow pointing in the direction of the flow at that point. This collection of arrows is a vector field: one vector for each point. Depending on how many fish you had, you could get a better (finer) or worse (coarser) visual representation of the actual flow of the water. Since the water itself is transparent, these arrows would give you a better idea of what’s actually taking place in the flow within the tank. Although it is physically impossible to achieve it, a vector field with vectors at each of the infinitely many points in the tank would give you complete information about the flow. Even though we can’t do this practically, mathematicians nevertheless can envisage and represent such vector fields. How do they do this?

Well, it is standard to represent a point in space by its coordinates. To keep things simple, let’s say we are dealing just with a flat space, as we do in plane geometry. Think of it as a piece of the floor. Each point, as you learn in coordinate geometry in high school, has two coordinates, and is represented as (x-coordinate, y-coordinate). Similarly, a vector or arrow on the floor is represented by two coordinates: the first tells how far to the right (positive) or left (negative) the arrow points; the second tells how far up (positive) or down (negative) it points. So, the vector (1,1) points one unit to the right and one unit up, so it makes an upward 45 degree angle with the horizontal. So does the vector (2,2), but it’s twice as long. Thus, two coordinates are enough to describe a vector on the floor. So a vector field on the floor would draw, at each point P = (x,y), an arrow or vector at P. The vector’s coordinates (which give its magnitude and direction) depend on the point at which it is to be drawn, so depend on the coordinates of that point. Thus, the first coordinate of the vector (giving its left/right direction), which we’ll call it X, depends on both x and y; we express this by writing X = f(x,y) or X is a function of x and y. Similarly, the vector’s second coordinate (giving its up/down direction) also depends on x and y, so is also a function of x and y; we write: Y = g(x,y). So a vector field for the points on the floor is represented by two functions X = f(x,y) and Y = g(x,y). Mathematicians put them together into a single pair which they write:

(X,Y) = (f(x,y), g(x,y))For example: (X,Y) = (x² + y, 3x + 4y)This vector field draws the vector (x² + y, 3x + 4y) at the point (x,y). So, for example, it draws the vector (X,Y) = (7,18) at the point (x,y) = (2,3), since 2² + 3 = 7 and 3(2) + 4(3) = 18.

When Larry drops some ink into the water on Charlie’s floor, he is trying to visualize the vector field of the flow. Each droplet spreads out along the flowing water. The faster the water is flowing at the point where the droplet is placed, the more elongated the drop becomes. It moves in the direction of the vector at that point, and at a rate roughly proportional to the length of that vector, at least during a fixed time interval, say of about a second or so. The more drops of ink Larry squeezes out of his pen, the more vectors get drawn. He see that some of this flow is not flat along the floor, but is moving in a pattern suggesting that some vectors are pointing up: the water is oozing upward, so the leak is coming from below.

Mathematicians use computers to visualize vector fields given by equations like the one I wrote above. Here is an example (I’ll post some more when I get a chance):