I’m currently taking multivariable calculus and it’s the last week of class. That said, I need to find ways to study. I’m one of those that believes the best way to learn is by teaching, ergo I’m going to try and explain vector calculus on this blog. I don’t expect it to be pretty. If you don’t have a little background in calculus you probably will have no idea what I’m talking about. If you have a background of calculus, good luck making sense of what I’m talking about. I appreciate any clarification questions as they will force me to reinforce my understanding by explaining things better.
We’ll see if this lasts more than one post. Vector Fields after the jump.
Updated: 5:51pm CDT, update highlighted in yellow
Today, I’m starting with the basics of a vector field. For those of you not familiar with a vector, it is drawn as a line with an arrow on the end. The vector is composed of two parts, a size (or magnitude) and a direction. Let’s look at an example:
Note, the vector starts where the two axis meet. This is called standard position. These axis are arbitrarily called the “v” and “u” axis, but they could easily be the x- and y-axis. Since we’ll also be looking at things in 3-D (3-space), the vector isn’t always going to be in just 2-D. Just keep that in mind.
I’m going to skip the basics on vectors as that was covered a few weeks ago; instead, I’m skipping right to vector fields. Vector fields are collections of many vectors over a space. One example of this is shown below with ocean currents.
As you can see above, there are several vectors (arrows) of different colors. The colors in this case represent the different size of the vector. Here, the size of the vector represents the speed of the ocean current, red being very fast and blue being very slow. So as you can see, the water moves in a direction, with a speed, and when those two are combined, it creates a vector field.
In this example, we are looking at a 2-D vector field (we’re not paying attention to water rising or sinking, just north/south and east/west). In a vector field of this style, we can create a vector function. Since the ocean current would be way too complicated, I’ll use this simplified equation:
F(x,y)=P(x,y)i+Q(x,y)j or F(x,y)=<P(x,y),Q(x,y)>
For those of you not familiar with i or j, they represent basic directions. i is the east/west direction, with east being positive in our example. j is the north/south direction, with north being the positive direction in our example. That said, we can see what direction P and Q are oriented with. In this equation, P(x,y) is a complicated equation that describes the flow of ocean currents just in the east/west direction (x direction or i). The Q(x,y) is another complicated equation that describes the flow of ocean currents just in the north/south direction (y direction or j). The second part of the equation with the “<>”s is just an easier way to write the same thing as the left equation. I’ll be using it primarily.
One operation that you need to be accustomed to seeing is the gradient operator. The gradient is represented by the symbol ∇. This operator takes the derivative (see calculus one for this) of each part of a function with respect to its position. In other words, you take the derivative of the x or i component of the function F and place that as the new x value of a function f, and you would take the derivative of the y or j direction and place that as the new y value of the same function f. Let’s take a look at that equation and what it means:
∇ F(x,y) = f(x,y) = <Px(x,y),Qy(x,y)>
You’ll notice that the subscripts tell you what you are taking the derivative in respect to. UPDATE: Take note that the gradient of the vector field is yet another vector field. [end of update] Now the question comes up of what does this gradient operator give us? The answer to that takes some geometrical reasoning that I will try to explain as best I can. If you look closely at the ocean current vector field above, you’ll notice some lines in the background in gray. You can see that these lines also go parallel to the arrows around them. These lines are called level curves.
Level curves represent the lines of constant flow. The gradient function is perpendicular to these lines. The gradient gives us the fastest rate of change (remember from calculus I, derivative gives you a rate of change). In this case, the gradient gives you the rate of change of ocean current speed from one point to another.
Say you are at the Antarctic coast in the bottom left of the screen, and you want to get to an place in the ocean with quicker flow to move your boat around the world faster. If you move directly east or west, the change in speed will not be that much. If you move directly north, you can see the ocean current flow is much faster, BUT as you can see the level curve is slightly angled to the southeast direction. By taking the gradient of the ocean current function (F), you will find what direction will allow you to change the quickest. So instead of heading directly north to get to the quicker ocean flows, you can minimize the distance by traveling in a particular direction slightly north by northeast.
When you get to 3-D, the idea remains similar, but is much more complicated.
This will be all for this individual lesson. We’re working in baby steps here. The next lesson should be about conservative vector fields along with curl and divergence.