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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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# Green Bean Matherole

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林宜悉 posted on 2020/03/30
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