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.