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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.

OK, I know some people aren't into green bean casseroles,

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B2 casserole vector bean field string wind

Green Bean Matherole

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    林宜悉 posted on 2020/03/30
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