Roger's Equations

In Vectors and Scalars - Part I, I discussed the difference between Vectors and Scalars, the allowed operations between the two, which coordinate systems work best for each, and presented the idea of both scalar and vector fields.

In the equation below:

Force (a vector field), is equal to the negative of the gradient () of the potential (a scalar field). The symbol is called "del" and is defined to be:

Where

Are unit vectors. So the gradient measures the rate of change in each of the directions of your coordinate system. The gradient is called an operator because it is meaningless by itself. Only when it acts on something like a Scalar Field or a Vector Field is it meaningful.

In Part I, I talked about the type of operations allowed for Vectors and Scalars. Although the gradient is an operator, it interacts like a Vector. So the possible interactions are:

Vector x Scalar = Vector; Called "The Gradient"

Please note that the unit vectors i,j,k mean the same thing as the unit vectors x,y,z. Its just the math people like to use i,j,k to be more general. The point is that the result here is a vector.

Vector • Vector = Scalar ; Called "The Divergence"

Notice there are no unit vectors in the expression above. The result will be a Scalar, not a Vector.

Vector x Vector = Vector ; Called "The Curl"

The last expression is complicated, involving a determinate. The important thing to take away is that the result is a Vector (see the ijk in the determinate, right?).

So clearly the operator "del" can be used on Vectors as Well as Scalars in very specific ways. So what does it tell us? Well, luckily the names of each of the operations above give good clues to what they tell us. Before we get into all that, lets get basic:

This is the derivative with respect to x of a function. The function can be a Vector or Scalar function. Basically, it determines the change in the function for a small change in x. In the vector case it measures a small change in the function for a small change in x in the x-direction. We can call it the rate of change.

So lets consider the gradient. The gradient basically takes the rate of change of the function at a point and assigns a vector to it. The magnitude of the vector indicates the magnitude of the rate of change, the direction of the vector indicates the direction in which the rate of change increases. Consider the picture below;

Lets say there is a heater in the room and temperature difference is indicated by color. In the picture on the left, the heater is in the center of the room and in the picture on the right, the heater is along the lefthand side. As we get further from the heater, the temperature gets lower, quickly at first, then more slowly.

From Part I we know that temperature is a Scalar Field, so what are the arrows? The arrows represent the Gradient of the Scalar Field, which is a Vector Field. Notice the arrows point in the direction of greatest rate change.

Next lets look at the Divergence.

Lets say that the figure above represents water flowing outward (Mayber the faucet is turned on and the water hits the bottom of a big bucket and spreads out. Notice that the vectors (which represent water velocity and direction) are spreading out as they get away from the source (faucet). If we'd like to know the rate with which these vectors are moving away from each other in the vector field, we'd take the divergence of that vector field.

Now for Curl:

Lets say we've filled our bucket up and turned the faucet off from our previous example. Now we stir the bucket. The velocity vector field would look something like the image above. Notice that each vector is rotated or curved slightly from the one behind it, leading to the water traveling in a circle. If we wanted to measure the rate of curving as well as the direct in which its curving towards, we'd take the Curl of the Vector Field, which would give us a new Vector Field.

So thats what Gradient, Divergence, and Curl tell us. Its all about the rate of change of a function in terms of values, spreading out, or curving. Obviously a Scalar function can't spread out (it has no direction) so the Curl and Divergence of a Scalar fuction is zero. The Curl of a vector field like the faucet example above would be equal to zero because there is no curving going on. The Divergence of a vector field like the stirring bucket example would be zero because there is no spreading out going on.

So why go through all this. How is this useful? Well, as I hinted at in the begining and you probably know, the gradient, divergence, and curl show up all the time in engineering. Lets take our original equation:

The Force is equal to the negative of the gradient of the potential energy. Remembering that gradient of the potential is simply a measure in the rate of change of the potential, this just says that a Force is equal to a negative rate of change in the potential energy function.

Consider one of Maxwell's Equations below,

It says the Divergence of the Magnetic field is zero. Remember, divergence tells us how much a Vector field "spreads out". So a Magnetic field doesn't spreads out. This is because there is no such thing as a Magnetic monopole. A bar magnet has a north and south, but you'll never find a particle that is just north or just south. No matter how small you go, if its generating a magnetic field, it will always look like a bar magnet, with a north and south.

Electric fields however...

...do spread out. Above, D is the Electric Field (a vector field) and ρ is charge density, which for a point charge is equal to the charge q. So things like Electrons and Protons, with charge -q and +q are monopoles, that is they are either positive or negative, but never both.

There are many other physical and engineering examples, but I think you get the picture. It is useful to describe quantities in science as Scalars and Vectors. Furthermore, gradients, divergences, and curls are useful in describing how different physical and engineering quantities are related.

One Additional Thought: The Laplacian

Earlier I said "del" (), was an operator that acted like a vector. What would happen if we took the dot product of two dels?:

Above you can see that what you get is another operator, one that measures the rate of diverging of the rate of change vector field, which we call the Laplacian. The Laplacian operator behaves like a scalar, as we would expect since:

Vector • Vector = Scalar

What if we took the Cross product of two dels ()?

xx F = (• F) - •(F) = (• F)-²F

In the equation above, F is just a vector function. Basically the above (which doen't have a fancy name as far as I know) says that the curl of the curl of a vector function produces a vector function which is the rate of change of the divergence minus the divergence of the rate of change. Yikes. Still this is used to derive the speed of light , so its useful.

If we include the function we can see some other interesting properties, for instance;

Curl of the Divergence

• (x F) = 0

So the curl of a vector function is a vector function that doesn't diverge (it curls in circles).

Curl of the Gradient

x (F) = 0

So the Gradient of a scalar function is a vector field that does't curve.

Well, that's enough. I hope I shed some insight into Vectors, Scalars, Fields, Functions, Gradients, Divergences, and Curls. Many of the relations in physics and engineering are related to rate of change and fields, so it makes sense these subjects would play a prominent role. As always, if you catch any mistakes, or have any suggestions for future derivations or discussions, please comment or email.

As always, I'd like to thank Wikipedia, along with Mathworld and Hyperphysics for images and some equations.