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OK, I know some people aren't into green bean casseroles, but I like them. Plus, they remind me of vector fields. Each green bean is like a little arrow, and I just have the urge to line them all up so they flow in the same direction. Maybe a little wavy, representing the vectors of flow in a river or something. Maybe complete little eddies. Or maybe the beans could represent wind vectors. Long beans would be high-magnitude vectors saying there's a strong wind in that direction, and short, low-magnitude beans would mean low wind speed. You could have a hurricane in your casserole dish with the long beans of high wind speed flowing counterclockwise near the center, mellowing out towards the outer edges of the storm. The center would have the shortest beans of all, showing the calm eye of the storm. Oh, and if you're wondering why I'm not curving the beans like this is because while vector fields might have a shape or flow to them, the vectors themselves don't. They're usually shown as straight lines, or numbers, or both. But that's not because they are straight lines. Vectors just represent what's happening at a single point. It's like this tiny point and this bit of wind can only travel in one direction at a time, so the bean points in that direction. And that tiny bit of wind has a certain speed, which is represented by the length of the bean. But the bean itself is just notation. Vectors themselves don't have a shape, just a direction and a magnitude, which means a bean with a direction and magnitude is just as legitimate a vector as an arrow plotted on a graph, or as a set of two numbers, or as one complex number, or as an orange slice cut with a certain angle and thickness, or as shouting a compass direction at a precise decibel level. North. East. I'll admit I'm not a huge fan of individual vectors sitting by themselves without meaning or context. One string bean does not make a casserole or matherole, as the case may be. But fields of vectors are awesome. They do have curves and patterns, context, and real-world meaning. There are vectorizable fields permeating this casserole dish right now-- the gravitational field, for instance. Gravitational forces are affecting all of my string beans, pulling them down towards the earth. And so you could use the string beans to create a vector-field casserole that actually represents the gravitational field they are currently in. Of course, this means just lining up the beans so all point down. And since they're all affected by basically the same amount of gravity, they should all be the same length. If you are cooking at a high altitude, be sure to cut your string beans shorter by an negligible amount. Another favorite vectorizable field of mine is also currently permeating these string beans-- the electromagnetic field. And if I had a giant bar magnet as a coaster-trivet thing, maybe I'd want my casserole to show the magnetic field that is actually there. The points near the poles of the magnet would have larger vectors, and they'd curve around just like iron filings do when you put them in a magnetic field. And the beans would show how the force weakens as it gets further from the magnet and goes from north to south. Or if you want to be true to life and don't have a magnet, you could put equal-sized string beans all pointing the same way, and then make sure your casserole is always pointing north, which might make it difficult to pass around the table, but I think dish-passing simplicity can be sacrificed for the sake of science, or mathematics, whatever this is. Speaking of which, you can also invent your own vector field by making up a rule for what the vector will be at each point. Like if you just said for any point you choose, you'll take the coordinates x comma y, and give that point a vector that's y comma x, so that this point, 0, 5, has the vector 5, 0. And at negative 3, negative 1, you have negative 1, negative 3. And negative 4, 4 gets 4 and negative 4. It's so simple. But you get this awesome vector field where the vectors kind of whoosh in from the corners and crash and whoosh out. Anyway, there's lots of other stuff you can do, but I'm going to go ahead and pour some goopy stuff into here and get this thing casserole-ing. It may not look very inspiring yet, but it's far from done. The most essential part of a matherole is an awesome oniony topping, and I've got just the trick. I will even show it to you in the next video.