Representing points in 3d

WEBVTT
Kind: captions
Language: en

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- [Voiceover] So a lot of the ways that we

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represent multivariable functions

00:00:04.041 --> 00:00:05.875
assume that you're
fluent with understanding

00:00:05.875 --> 00:00:08.522
how to represent points
in three-dimensions

00:00:08.522 --> 00:00:11.100
and also how to represent
vectors in three-dimensions.

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So I thought I'd make
a little video here to,

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spell out exactly how
it is that we describe

00:00:15.279 --> 00:00:17.369
points and vectors in three-dimensions.

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And before we do that,

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I think it will be valuable

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if we start off by describing

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points and vectors in two-dimensions.

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And, I'm assuming if you're learning

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about multivariable calculus,

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that a lot of you have
already learned about this.

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And you might be saying

00:00:30.396 --> 00:00:31.417
what's the point?

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I already know how to represent,

00:00:32.926 --> 00:00:36.015
points and vectors in two-dimensions.

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But there is a huge value in analogy here,

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because as soon as you start to compare

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two dimensions and three-dimensions,

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you start to see patterns for how it could

00:00:46.115 --> 00:00:47.996
extend to other dimensions that you

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can't necessarily visualize,

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or when it might be useful to think about

00:00:51.433 --> 00:00:53.244
one dimension versus another.

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So in two-dimensions, if
you have some kind of point

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just, you know, off sitting there.

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We typically represent it,

00:00:59.931 --> 00:01:01.998
you've got an x-axis and a y-axis

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that a perpendicular to each other.

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And we represent this number with a pair.

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Sorry, we represent this
point with a pair of numbers.

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So in this case, I don't know,

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it might be something like one, three.

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And what that would represent,

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is it's saying that you
have to move a distance

00:01:16.685 --> 00:01:19.185
of one along the x-axis

00:01:20.261 --> 00:01:24.139
and then a distance of
three up along the y-axis.

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So you know this, let's say
that's a distance of one,

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that's a distance of three,

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it might not be exactly
that the way I drew it,

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but let's say that those
are the coordinates.

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What this means is that every

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point in two dimensional space

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can be given a pair of numbers like this

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and you think of them as instructions

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where it's kind of telling you

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how far to walk in one way,

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how far to walk in another.

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But you can also think
of the reverse, right?

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Every time that you have a pair of things

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you know that you should be
thinking two-dimensionaly

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and that's actually,

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a surprisingly powerful idea that

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I don't think I
appreciated for a long time

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how there's this back and forth between

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pairs of numbers and points in space

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and it lets you visualize things

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you didn't think you could visualize,

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or lets you understand things

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that are inherently visual just by

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kinda going back and forth.

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And in three-two-dimensions

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there's a similar dichotomy, but between

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triplets of points and

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points in three-dimensional space.

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So, let me just plop down a point

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in three-dimensional space here,

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and it's hard to get a feel for exactly

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where it is until you move things around.

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This is one thing that
makes three-dimensions hard

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is you can't really draw it without

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moving it around or showing,

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showing a difference in
perspective in various ways.

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But we describe points like this,

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again with a set of coordinates,

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but this time it's a triplet.

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And this particular point,

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I happen to know is one, two, five.

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And what those numbers are telling you is

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how far to move, parallel to each axis.

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So just like with two dimensions,

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we have an x-axis,

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and a y-axis.

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But now there's a third axis that's

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perpendicular to both of them,

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and moves us into that
third-dimension, the z-axis.

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And the first number in our coordinate

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is gonna tell us how far,

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whoop, can't move those guys,

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how far we need to move in the x direction

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as our first step.

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The second number, two in this case,

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tells us how far we need to move

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parallel to the y-axis,
for our second step.

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And then the third number
tells us how far up

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we have to go to get to that point.

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And you can do this for any point

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in three-dimensional space, right?

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Any point that you have you can

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give the instructions for how to move

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along the x, and then how
to move parallel to the y

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and how to move parallel to the z

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to get to that point,

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which means there's this back and forth

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between triplets of
numbers and points in 3-D.

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So whenever you come
across a triplet of things,

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and you'll see this in the next video

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when we start talking about
three-dimensional graphs,

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you'll know, just by virtue of the

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fact that it's a triplet

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"Ah, yes, I should be thinking
in three-dimensions somehow"

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just in the same way
whenever you have pairs

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you should be thinking

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ah, this is a very two-dimensional thing.

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So, there's another context though

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where pairs of numbers come up

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and that would be vectors.

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So a vector you might represent,

00:04:03.427 --> 00:04:05.958
you know you typically it with an arrow.

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Oh,

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ahh.

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Help, help! (chuckling)

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So vectors,

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So vectors we typically
represent some kind of arrow,

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let's, this arrow nice color.

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An arrow.

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And if it's a vector from
the origin to a simple point,

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the coordinates of that vector

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are just the same as those of the point.

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And the convention is to write

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those coordinates in a column.

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You know, it's not set in stone,

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but typically if you
see numbers in a column

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you should be thinking
about it as a vector,

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some kind of arrow.

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And if it's a pair with
parenthesis around it

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you just think about it as a point.

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And even though, you know, both of these

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are ways of representing
the same pair of numbers,

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the main difference is that a vector

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you could have started
at any point in space

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it didn't have to start in the origin.

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So if we have that same guy,

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but you know if he starts here

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and he still has a
rightward component of one

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and an upward component of three,

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we think of that as the same vector.

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And typically these are
representing motion of some kind

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whereas points are just representing

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like actual points in space.

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And the other big thing that you can do

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is you can add vectors together.

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So, you know, if you had another,

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let's say you have another vector

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that has a large x component

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but a small negative y
component, like this guy.

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And what that means is
that you can kind of add

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like, imagining that second vector started

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at the tip of the first one

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and then however you get from the origin

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to the new tip there,

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that's gonna be the resulting vector, so

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I'd say this is, this is the
sum of those two vectors.

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And you can't really do
that with points as much.

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In order to think about adding points

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you end up thinking about them as vectors.

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And the same goes with three-dimensions.

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For a given point, if you draw an arrow

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from the origin up to that point,

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this arrow would be represented

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with that same triplet of numbers,

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but you typically do it in a column,

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I call this a column vector.

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That's not three, that's five.

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And the difference between the point

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and the arrow is you can think of,

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you know the arrow or the vector

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is starting anywhere in space,

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it doesn't really matter,

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as long as it's got those same components

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for how far does it
move parallel to the x,

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for how far does it move
parallel to the y-axis,

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and for how far does it
move parallel to the z-axis.

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So in the next video,

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I'll show how we use
these three-dimensions

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to start graphing multivariable functions.


Parametric curves

WEBVTT
Kind: captions
Language: en

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- [Voiceover] More
function visualizations.

00:00:02.711 --> 00:00:04.644
So let's say you have a function

00:00:04.644 --> 00:00:06.831
that's got a single input t.

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And then it outputs a vector.

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And the vector's gonna depend on t.

00:00:11.869 --> 00:00:16.136
So the x component will be
t times the cosine of t.

00:00:17.216 --> 00:00:21.906
And then the y component will
be t times the sine of t.

00:00:23.216 --> 00:00:25.530
This is what's called
a parametric function.

00:00:26.520 --> 00:00:30.119
And I should maybe say
one-parameter parametric function.

00:00:30.119 --> 00:00:31.134
One parameter.

00:00:31.134 --> 00:00:34.695
And "parameter" is just kind
of a fancy word for input.

00:00:34.695 --> 00:00:36.374
Parameter.

00:00:37.326 --> 00:00:40.562
So in this case, t is
our single parameter.

00:00:40.562 --> 00:00:42.157
And what makes it a parametric function

00:00:42.157 --> 00:00:44.834
is that we think about
it as drawing a curve

00:00:44.834 --> 00:00:47.143
and its output is multidimensional.

00:00:47.143 --> 00:00:49.590
So you might think, when you
visualize something like this,

00:00:49.590 --> 00:00:51.297
ah, it's got, you know, a single input.

00:00:51.297 --> 00:00:53.233
And it's got a two-dimensional output.

00:00:53.233 --> 00:00:54.080
Let's graph it.

00:00:54.080 --> 00:00:57.132
You know, put those three
numbers together and plot them.

00:00:57.132 --> 00:00:58.729
But what turns out to be even better

00:00:58.729 --> 00:01:01.508
is to look just in the output space.

00:01:01.508 --> 00:01:03.714
So in this case, the output
space is two-dimensional.

00:01:03.714 --> 00:01:07.078
So I'll go ahead and draw
a coordinate plane here.

00:01:07.078 --> 00:01:09.622
And let's just evaluate this function

00:01:09.622 --> 00:01:10.868
at a couple different points

00:01:10.868 --> 00:01:12.968
and see what it looks like, okay.

00:01:12.968 --> 00:01:15.028
So I might, maybe the easiest place

00:01:15.028 --> 00:01:16.539
to evaluate it would be zero.

00:01:18.049 --> 00:01:20.186
So f of zero

00:01:23.766 --> 00:01:24.612
is equal to,

00:01:24.612 --> 00:01:27.665
and then, in both cases,
it'll be zero times something.

00:01:27.665 --> 00:01:30.891
So zero times cosine of zero is just zero.

00:01:30.891 --> 00:01:33.882
And then zero times sine
of zero is also just zero.

00:01:35.532 --> 00:01:37.396
So that input corresponds to the output.

00:01:37.396 --> 00:01:39.653
You could think of it as a
vector that's infinitely small

00:01:39.653 --> 00:01:41.974
or just the point at the origin,

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however you want to go about it.

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So let's take a different point,

00:01:45.056 --> 00:01:47.419
just to see what else could happen.

00:01:48.989 --> 00:01:51.013
And I'm gonna choose pi halves.

00:01:52.873 --> 00:01:55.064
Of course, the reason
I'm choosing pi halves,

00:01:55.064 --> 00:01:57.312
of all numbers, is that
it's something I know

00:01:57.312 --> 00:01:59.097
how to take the sine and the cosine of.

00:02:00.447 --> 00:02:03.426
So t is pi halves, cosine of pi halves.

00:02:05.206 --> 00:02:06.041
And you start thinking, okay,

00:02:06.041 --> 00:02:07.273
what's the cosine of pi halves?

00:02:07.273 --> 00:02:08.663
What's the sine of pi halves?

00:02:08.663 --> 00:02:11.458
And maybe you go off and
draw a little unit circle

00:02:11.458 --> 00:02:14.203
while you're writing things out.

00:02:14.203 --> 00:02:14.902
Whoops.

00:02:16.882 --> 00:02:18.732
See the problem with
talking while writing.

00:02:19.602 --> 00:02:20.838
Sine of pi halves.

00:02:24.498 --> 00:02:26.303
And you know it's,

00:02:26.303 --> 00:02:28.841
if you go off and scribble
that little unit circle.

00:02:28.841 --> 00:02:30.574
And you say pi halves is gonna bring us

00:02:30.574 --> 00:02:32.013
a quarter of the way around.

00:02:32.013 --> 00:02:33.179
Over here.

00:02:33.179 --> 00:02:36.205
And cosine of pi halves is measuring

00:02:36.205 --> 00:02:37.542
the x component of that.

00:02:37.542 --> 00:02:39.320
So that just cancels out to zero.

00:02:39.320 --> 00:02:43.308
And then sine of pi halves

00:02:43.308 --> 00:02:45.004
is the y component of that.

00:02:45.004 --> 00:02:46.723
So that ends up equaling one.

00:02:50.093 --> 00:02:51.832
Which means that the vector as a whole

00:02:55.132 --> 00:02:57.744
is gonna be zero for the x component

00:02:57.744 --> 00:02:59.796
and then pi halves for the y component.

00:03:01.286 --> 00:03:02.407
And what that would look like,

00:03:02.407 --> 00:03:04.973
you know, the y component of pi halves

00:03:04.973 --> 00:03:07.183
is about 1.7 up there.

00:03:07.183 --> 00:03:08.508
There's no x component.

00:03:08.508 --> 00:03:10.011
So you might get a vector like this.

00:03:10.011 --> 00:03:11.550
And if you imagine doing this

00:03:11.550 --> 00:03:13.124
at all the different input points,

00:03:13.124 --> 00:03:15.210
you might get a bunch of different vectors

00:03:15.210 --> 00:03:17.060
off doing different things.

00:03:17.060 --> 00:03:18.468
And if you were to draw it,

00:03:18.468 --> 00:03:21.080
you don't want to just
draw the arrows themselves.

00:03:21.080 --> 00:03:22.749
Because that'll clutter
things a whole bunch.

00:03:22.749 --> 00:03:24.725
So we just want to trace the points

00:03:24.725 --> 00:03:27.219
that correspond to the output,
the tips of each vector.

00:03:27.219 --> 00:03:29.741
And what I'll do here, I'll
show a little animation.

00:03:29.741 --> 00:03:31.956
Let me just clear the board a bit.

00:03:32.796 --> 00:03:34.153
An animation where

00:03:34.153 --> 00:03:36.680
I'll let t range between zero and 10.

00:03:37.690 --> 00:03:39.323
So let's write that down.

00:03:43.024 --> 00:03:44.973
So the value t is gonna start at zero.

00:03:44.973 --> 00:03:47.411
And then it's gonna go to 10.

00:03:47.411 --> 00:03:50.202
And we'll just see what values,

00:03:50.202 --> 00:03:52.171
what vectors does that output.

00:03:52.171 --> 00:03:55.483
And what curve does the tip
of that vector trace out?

00:03:57.433 --> 00:03:58.813
So there it goes.

00:03:58.813 --> 00:04:01.199
All of the values just kind
of ranging, zero to 10.

00:04:01.199 --> 00:04:03.453
And you end up getting this spiral shape.

00:04:03.453 --> 00:04:05.099
And you can maybe think about why

00:04:05.099 --> 00:04:08.272
this cosine of t, sine of t
scaled by the value t itself

00:04:08.272 --> 00:04:09.542
would give you this spiral.

00:04:09.542 --> 00:04:11.576
But what it means is that when you,

00:04:12.976 --> 00:04:14.948
you know, we saw that zero goes here.

00:04:14.948 --> 00:04:19.591
Evidently, it's the case
that 10 outputs here.

00:04:19.591 --> 00:04:21.578
And a disadvantage of
drawing things like this,

00:04:21.578 --> 00:04:24.619
you're not quite sure of
what the interim values are.

00:04:24.619 --> 00:04:25.731
You know, you could kind of guess.

00:04:25.731 --> 00:04:27.230
Maybe one goes somewhere here.

00:04:27.230 --> 00:04:28.801
Two goes somewhere here.

00:04:28.801 --> 00:04:31.144
And you're kind of hoping
that they're evenly spaced

00:04:31.144 --> 00:04:32.441
as you move along.

00:04:32.441 --> 00:04:34.638
But you don't get that information.

00:04:34.638 --> 00:04:36.388
You lose the input information.

00:04:36.388 --> 00:04:37.704
You get the shape of the curve.

00:04:37.704 --> 00:04:39.122
And if you just want, you know,

00:04:39.122 --> 00:04:40.953
an analytical way of describing curves,

00:04:40.953 --> 00:04:43.025
you find some parametric
function that does it.

00:04:43.025 --> 00:04:44.979
And you don't really care about the rate.

00:04:44.979 --> 00:04:46.577
But just to show where it might matter,

00:04:46.577 --> 00:04:49.482
I'll animate the same thing again,

00:04:49.482 --> 00:04:51.332
another function that
draws the same curve.

00:04:51.332 --> 00:04:53.236
But it starts going really quickly.

00:04:53.236 --> 00:04:55.577
And then it slows down as you go on.

00:04:56.497 --> 00:04:59.945
So that function is not quite
the t cosine t, t sine of t

00:04:59.945 --> 00:05:01.482
that I originally had written.

00:05:01.482 --> 00:05:02.972
And in fact, it would mean that,

00:05:03.942 --> 00:05:06.239
let's just erase these guys.

00:05:07.669 --> 00:05:08.728
When it starts slowly,

00:05:08.728 --> 00:05:11.298
you can interpret that
as saying, okay maybe...

00:05:11.298 --> 00:05:12.795
Well, actually, it started
quickly, didn't it?

00:05:12.795 --> 00:05:15.598
So one would be really far off here.

00:05:15.598 --> 00:05:17.340
And then two, we kind
of zipped along here.

00:05:17.340 --> 00:05:19.517
And three, you know,
still going really fast.

00:05:19.517 --> 00:05:21.316
And then maybe by the
time you get to the end,

00:05:21.316 --> 00:05:22.773
it's just going very slowly,

00:05:22.773 --> 00:05:24.925
just kind of seven is here, eight is here,

00:05:24.925 --> 00:05:27.801
and it's hardly making any
progress before it gets to 10.

00:05:27.801 --> 00:05:29.660
So you can have two different functions

00:05:29.660 --> 00:05:30.990
draw the same curve.

00:05:30.990 --> 00:05:33.670
And the fancy word here is "parameterize."

00:05:33.670 --> 00:05:36.363
So functions will parameterize a curve

00:05:36.363 --> 00:05:38.898
if, when you draw just
in the output space,

00:05:38.898 --> 00:05:40.411
you get that curve.

00:05:40.411 --> 00:05:41.455
And in the next video,

00:05:41.455 --> 00:05:43.307
I'll show how you can have functions

00:05:43.307 --> 00:05:44.668
with a two-dimensional input

00:05:44.668 --> 00:05:46.227
and a three-dimensional output

00:05:46.227 --> 00:05:49.430
draw surfaces in three-dimensional space.


3d vector field example

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:00.828
- [Voiceover] So in the last video,

00:00:00.828 --> 00:00:03.348
I talked about
3-dimensional vector fields.

00:00:03.348 --> 00:00:04.149
And I finished things off

00:00:04.149 --> 00:00:06.250
with a sort of identity function example

00:00:06.250 --> 00:00:09.803
where at an input point x, y, z

00:00:09.803 --> 00:00:12.485
the output vector is also x, y, z.

00:00:12.485 --> 00:00:13.924
And here, I want to go through

00:00:13.924 --> 00:00:16.596
a slightly more intricate example.

00:00:16.596 --> 00:00:19.776
So I'll go ahead and get
rid of this vector field

00:00:19.776 --> 00:00:21.540
and in this example,

00:00:21.540 --> 00:00:26.521
the x component of the
output will be y times z.

00:00:26.521 --> 00:00:30.225
The y component of the
output will be x times z.

00:00:30.225 --> 00:00:34.625
And the z component of the
output will be x times y.

00:00:34.625 --> 00:00:36.227
So I'll just show this vector field

00:00:36.227 --> 00:00:38.914
and then we can start to get a feel for

00:00:38.914 --> 00:00:40.743
how the function that I just wrote

00:00:40.743 --> 00:00:42.868
relates to the vectors that you're seeing.

00:00:42.868 --> 00:00:43.843
So you see some of the vectors

00:00:43.843 --> 00:00:45.701
are kind of pointing away from the origin.

00:00:45.701 --> 00:00:48.209
Some are pointing in towards the origin.

00:00:50.779 --> 00:00:52.353
So how can we understand this vector field

00:00:52.353 --> 00:00:54.478
in terms of the function itself?

00:00:54.478 --> 00:00:55.976
And a good step is to just zero in

00:00:55.976 --> 00:00:57.473
on one of the components.

00:00:57.473 --> 00:00:59.377
So in this case, I'll
choose the z component

00:00:59.377 --> 00:01:00.968
of the output

00:01:00.968 --> 00:01:03.035
which is made up of x times y

00:01:03.035 --> 00:01:03.975
and kind of start to understand

00:01:03.975 --> 00:01:04.869
what that should be.

00:01:04.869 --> 00:01:06.552
The z component will represent

00:01:06.552 --> 00:01:08.851
how much the vector's pointing up or down.

00:01:08.851 --> 00:01:10.256
This is the z axis

00:01:10.256 --> 00:01:11.835
and the x-y plane here

00:01:11.835 --> 00:01:13.890
so I'll point the z axis
straight at our face.

00:01:13.890 --> 00:01:16.374
This is the x axis, this is the y axis.

00:01:16.374 --> 00:01:18.859
The values of x and y are
going to completely determine

00:01:18.859 --> 00:01:20.136
that z component.

00:01:20.136 --> 00:01:21.378
So I'm going to go off to the side here

00:01:21.378 --> 00:01:25.976
and just draw myself a little
x-y plane for reference.

00:01:25.976 --> 00:01:28.878
So this is my x value,

00:01:28.878 --> 00:01:30.353
this is my y value.

00:01:30.353 --> 00:01:31.688
And I want to understand the meaning

00:01:31.688 --> 00:01:35.008
of the term "x times y."

00:01:35.008 --> 00:01:37.412
So when both x and y are positive,

00:01:37.412 --> 00:01:39.617
the product is positive.

00:01:39.617 --> 00:01:40.976
And when both of them are negative,

00:01:40.976 --> 00:01:43.449
the product is also positive.

00:01:43.449 --> 00:01:45.469
If x is negative and y is positive,

00:01:45.469 --> 00:01:46.874
the product is negative,

00:01:46.874 --> 00:01:48.778
but if x is positive and y is negative,

00:01:48.778 --> 00:01:50.670
the product is also negative.

00:01:50.670 --> 00:01:51.436
So what this should mean,

00:01:51.436 --> 00:01:52.772
in terms of our vector field,

00:01:52.772 --> 00:01:54.292
is that when we're in this first quadrant,

00:01:54.292 --> 00:01:56.858
vectors tend to point
up in the z direction,

00:01:56.858 --> 00:01:59.419
same over here in the third quadrant.

00:01:59.419 --> 00:01:59.970
But over in the other two

00:01:59.970 --> 00:02:01.479
they should tend to point down.

00:02:01.479 --> 00:02:02.895
So let's focus in on that first quadrant

00:02:02.895 --> 00:02:04.614
and try to look at what's going on.

00:02:04.614 --> 00:02:07.075
So, you see like this vector
here applies, this vector.

00:02:07.075 --> 00:02:09.269
And all of them generally point upwards,

00:02:09.269 --> 00:02:11.139
they have a positive z component.

00:02:11.139 --> 00:02:13.054
So that seems in line with
what we were predicting.

00:02:13.054 --> 00:02:13.960
Whereas over here,

00:02:13.960 --> 00:02:15.574
which corresponds to the fourth quadrant

00:02:15.574 --> 00:02:16.653
of the x-y plane,

00:02:16.653 --> 00:02:19.835
the z component of each
vector tends to be down.

00:02:19.835 --> 00:02:22.040
And they're doing other things

00:02:22.040 --> 00:02:23.515
in terms of the x and y components,

00:02:23.515 --> 00:02:24.920
it's not just z component action,

00:02:24.920 --> 00:02:26.940
but right now we're just
focusing on up and down.

00:02:26.940 --> 00:02:29.668
And if you look over
in the third quadrant,

00:02:29.668 --> 00:02:31.154
they tend to be pointing up,

00:02:31.154 --> 00:02:32.948
and that corresponds to the fact

00:02:32.948 --> 00:02:34.510
that x times y will be positive.

00:02:34.510 --> 00:02:35.102
And when you look at it,

00:02:35.102 --> 00:02:36.785
it all starts to align that way.

00:02:36.785 --> 00:02:38.329
And because I chose a
rather symmetric function

00:02:38.329 --> 00:02:40.047
you could imagine doing this where you,

00:02:40.047 --> 00:02:43.182
you analyze also the y component here,

00:02:43.182 --> 00:02:46.224
you analyze the x component
in terms of y and z,

00:02:46.224 --> 00:02:47.582
and it's actually going
to look very similar

00:02:47.582 --> 00:02:50.090
for understanding when the
x component of a vector

00:02:50.090 --> 00:02:51.228
tends to be positive,

00:02:51.228 --> 00:02:51.924
like up here,

00:02:51.924 --> 00:02:53.085
or when it turns out to be negative,

00:02:53.085 --> 00:02:54.339
like over here,

00:02:54.339 --> 00:02:55.663
and same with when the y
component of the vector

00:02:55.663 --> 00:02:56.789
tends to be negative,

00:02:56.789 --> 00:02:58.949
or if the y component
tends to be positive.

00:02:58.949 --> 00:03:02.420
And overall, it's a very
complicated image to look at,

00:03:02.420 --> 00:03:05.566
but you can slowly piece
by piece get a feel for it.

00:03:05.566 --> 00:03:07.203
And just like with
2-dimensional vector fields,

00:03:07.203 --> 00:03:08.387
the kind of neat thing to do

00:03:08.387 --> 00:03:11.046
is imagine that this
represents a fluid flow.

00:03:11.046 --> 00:03:12.822
So you imagine like maybe air around you,

00:03:12.822 --> 00:03:14.935
flowing in towards the origin here,

00:03:14.935 --> 00:03:17.048
flowing out away from the origin there,

00:03:17.048 --> 00:03:19.173
it would kind of be rotating around here,

00:03:19.173 --> 00:03:21.387
and later on in multi-variable calculus,

00:03:21.387 --> 00:03:23.399
you'll learn about
various ways you can study

00:03:23.399 --> 00:03:25.698
just the function itself
and just the variables

00:03:25.698 --> 00:03:28.519
and get a feel for how that
fluid itself would behave,

00:03:28.519 --> 00:03:30.725
even though it's a very
complicated thing to think about,

00:03:30.725 --> 00:03:31.967
it's even complicated to draw

00:03:31.967 --> 00:03:33.581
or use graphic software with,

00:03:33.581 --> 00:03:35.299
but just with analytic tools,

00:03:35.299 --> 00:03:38.762
you can get very powerful results.

00:03:38.762 --> 00:03:40.930
And these kind of things come
up in physics all the time

00:03:40.930 --> 00:03:42.567
because you're thinking
in 3-dimensional space

00:03:42.567 --> 00:03:44.587
and it doesn't just have to be fluid flow,

00:03:44.587 --> 00:03:47.188
it could be a force field
like an electric force field

00:03:47.188 --> 00:03:48.836
or gravitational force field,

00:03:48.836 --> 00:03:49.765
where each vector tells you

00:03:49.765 --> 00:03:52.145
how a particle tends to get pushed.

00:03:52.145 --> 00:03:54.328
And as we continue on with
multi-variable calculus,

00:03:54.328 --> 00:03:56.337
you'll get to see lots more examples.

00:03:56.337 --> 00:03:57.695
But hopefully this helps give a small feel

00:03:57.695 --> 00:03:59.053
for how you can go piece by piece

00:03:59.053 --> 00:04:00.470
and understand something

00:04:00.470 --> 00:04:03.149
that's kind of a complicated expression.


Contour plots

WEBVTT
Kind: captions
Language: en

00:00:01.048 --> 00:00:04.762
- [Voiceover] So I have here
a three-dimensional graph,

00:00:04.762 --> 00:00:05.999
and that means that it's representing

00:00:05.999 --> 00:00:07.545
some kind of function that has

00:00:07.545 --> 00:00:10.743
a two-dimensional input
and a one-dimension output.

00:00:10.743 --> 00:00:14.326
So that might look
something like f of x, y

00:00:15.489 --> 00:00:17.396
equals and then just some

00:00:17.396 --> 00:00:20.359
expression that has a
bunch of x's and y's in it.

00:00:20.359 --> 00:00:21.471
And graphs are great

00:00:21.471 --> 00:00:23.471
but they're kinda clunky to draw.

00:00:23.471 --> 00:00:25.395
I mean, certainly you can't
just scribble it down.

00:00:25.395 --> 00:00:28.240
It typically requires some
kind of graphing software

00:00:28.240 --> 00:00:29.702
and when you take a static image of it

00:00:29.702 --> 00:00:31.792
it's not always clear what's going on.

00:00:31.792 --> 00:00:33.743
So here I'm going to
describe a way that you can

00:00:33.743 --> 00:00:37.144
represent these functions and
these graphs two-dimensionally

00:00:37.144 --> 00:00:40.581
just by scribbling down on a
two-dimensional piece of paper.

00:00:40.581 --> 00:00:43.156
This is a very common way
that you'll see if you're

00:00:43.156 --> 00:00:46.085
reading a textbook or if someone
is drawing on a blackboard.

00:00:46.085 --> 00:00:48.209
It's known as a contour plot

00:00:48.209 --> 00:00:49.799
and the idea of a contour plot

00:00:49.799 --> 00:00:51.132
is that we're going to take this graph

00:00:51.132 --> 00:00:53.096
and slice it a bunch of times.

00:00:53.096 --> 00:00:55.060
So I'm going to slice
it with various planes

00:00:55.060 --> 00:00:58.903
that are all parallel to the x, y plane

00:00:58.903 --> 00:01:00.169
and let's think for a moment

00:01:00.169 --> 00:01:02.349
about what these guys represent.

00:01:02.349 --> 00:01:04.650
So the bottom one here

00:01:04.650 --> 00:01:08.650
represents the value z
is equal to negative two.

00:01:10.130 --> 00:01:11.781
This is the z-axis over here

00:01:11.781 --> 00:01:13.664
and when we fix that to be negative two

00:01:13.664 --> 00:01:16.940
and let x and y run freely
we get this whole plane.

00:01:16.940 --> 00:01:20.549
And if you let z increase,
keep it constant,

00:01:20.549 --> 00:01:22.504
but let it increase by one to negative one

00:01:22.504 --> 00:01:26.671
we get a new plane, still
parallel to the x, y plane

00:01:28.668 --> 00:01:31.318
but it's distance from the
x, y plane is negative one.

00:01:31.318 --> 00:01:32.173
And the rest of these guys,

00:01:32.173 --> 00:01:35.829
they're all still constant values of z.

00:01:35.829 --> 00:01:38.344
Now in terms of our graph,

00:01:38.344 --> 00:01:39.268
what that means is that

00:01:39.268 --> 00:01:44.175
these represent constant
values of the graph itself.

00:01:44.175 --> 00:01:47.185
These represent constant
values for the function itself.

00:01:47.185 --> 00:01:49.685
So because we always represent
the output of the function

00:01:49.685 --> 00:01:52.122
as the height off of the x, y plane

00:01:52.122 --> 00:01:56.039
these represent constant
values for the output.

00:01:58.330 --> 00:02:00.650
What that's going to look like.

00:02:00.650 --> 00:02:02.167
So what we do

00:02:02.167 --> 00:02:05.751
is we say, "Where do these
slices cut into the graph?"

00:02:05.751 --> 00:02:08.127
So I'm going to draw on all of the points

00:02:08.127 --> 00:02:09.916
where those slices cut into the graph

00:02:09.916 --> 00:02:11.889
and these are called contour lines.

00:02:11.889 --> 00:02:12.957
We're still in three-dimensions

00:02:12.957 --> 00:02:15.349
so we're not done yet.

00:02:15.349 --> 00:02:18.017
So what I'm going to do is
take all these contour lines

00:02:18.017 --> 00:02:21.456
and I'm going to squish them
down on to the x, y plane.

00:02:21.456 --> 00:02:23.638
So what that means,

00:02:23.638 --> 00:02:26.750
each of them has some kind
of z component at the moment,

00:02:26.750 --> 00:02:28.632
and we're just going to chop it down,

00:02:28.632 --> 00:02:31.132
squish them all nice and flat,

00:02:31.998 --> 00:02:34.204
on to the x, y plane.

00:02:34.204 --> 00:02:36.885
And now we have something two-dimensional,

00:02:36.885 --> 00:02:38.069
and it still represents

00:02:38.069 --> 00:02:39.614
some of the outputs of our function.

00:02:39.614 --> 00:02:41.041
Not all of them, it's not perfect,

00:02:41.041 --> 00:02:43.941
but it does give a very good idea.

00:02:43.941 --> 00:02:46.745
I'm going to switch over to
a two-dimensional graph here.

00:02:46.745 --> 00:02:48.845
And this is that same

00:02:48.845 --> 00:02:51.162
function that we were just looking at.

00:02:51.162 --> 00:02:55.042
Let's actually move it a
little bit more central here.

00:02:55.042 --> 00:02:56.400
So this is the same function

00:02:56.400 --> 00:02:59.150
that we were just looking at, but

00:03:00.584 --> 00:03:01.546
each of these lines represents

00:03:01.546 --> 00:03:03.490
a constant output of the function

00:03:03.490 --> 00:03:04.802
so it's important to realize

00:03:04.802 --> 00:03:07.208
we're still representing

00:03:07.208 --> 00:03:09.665
a function that has a
two-dimensional input

00:03:09.665 --> 00:03:11.051
and a one-dimensional output.

00:03:11.051 --> 00:03:13.237
It's just that we're
looking in the input space

00:03:13.237 --> 00:03:14.658
of that function as a whole.

00:03:14.658 --> 00:03:16.663
So this is still

00:03:16.663 --> 00:03:17.496
f of x, y

00:03:19.954 --> 00:03:22.655
and then some expression of those guys

00:03:22.655 --> 00:03:26.099
but this line might represent
the constant value of f

00:03:26.099 --> 00:03:29.559
when all of the values
were at outputs three.

00:03:29.559 --> 00:03:31.716
Over here, this also,

00:03:31.716 --> 00:03:33.239
both of these circles together give you

00:03:33.239 --> 00:03:35.771
all the values where f outputs three.

00:03:35.771 --> 00:03:37.917
This one over here

00:03:37.917 --> 00:03:39.355
will tell you where it outputs two

00:03:39.355 --> 00:03:41.665
and you can't know this just
looking at the contour plot

00:03:41.665 --> 00:03:43.459
so typically if someone's drawing it

00:03:43.459 --> 00:03:46.185
if it matters that you
know the specific values

00:03:46.185 --> 00:03:47.734
they'll mark it somehow.

00:03:47.734 --> 00:03:51.357
They'll let you know what
value each line corresponds to

00:03:51.357 --> 00:03:53.901
but as soon as you know that
this line corresponds to zero

00:03:53.901 --> 00:03:56.432
it tells you that every
possible input point

00:03:56.432 --> 00:03:58.157
that sits somewhere on this line

00:03:58.157 --> 00:04:01.583
will evaluate to zero when you
pump it through the function.

00:04:01.583 --> 00:04:03.049
And this actually gives a very good feel

00:04:03.049 --> 00:04:04.759
for the shape of things.

00:04:04.759 --> 00:04:06.474
If you like thinking in terms of graphs

00:04:06.474 --> 00:04:08.049
you can kind of imagine how

00:04:08.049 --> 00:04:11.107
these circles and everything
would pop out of the page.

00:04:11.107 --> 00:04:11.940
You can also look,

00:04:11.940 --> 00:04:14.659
notice how the lines are really
close together over here,

00:04:14.659 --> 00:04:15.637
very, very close together,

00:04:15.637 --> 00:04:18.034
but they're a little
more spaced over here.

00:04:18.034 --> 00:04:20.460
How do you interpret that?

00:04:20.460 --> 00:04:24.977
Well, over here this means it
takes a very, very small step

00:04:24.977 --> 00:04:27.717
to increase the value
of the function by one,

00:04:27.717 --> 00:04:29.865
very small step and it increases by one,

00:04:29.865 --> 00:04:32.082
but over here it takes a much larger step

00:04:32.082 --> 00:04:34.534
to increase the function
by the same value.

00:04:34.534 --> 00:04:36.935
So over here this kind of means steepness.

00:04:36.935 --> 00:04:40.328
If you see a very short
distance between contour lines

00:04:40.328 --> 00:04:41.527
it's going to be very steep

00:04:41.527 --> 00:04:44.181
but over here it's much more shallow.

00:04:44.181 --> 00:04:46.311
And you can do things like this
to kind of get a better feel

00:04:46.311 --> 00:04:47.885
for the function as whole.

00:04:47.885 --> 00:04:49.846
The idea of a whole bunch
of concentric circles

00:04:49.846 --> 00:04:53.689
usually corresponds to
a maximum or a minimum,

00:04:53.689 --> 00:04:55.219
and you end up seeing these a lot.

00:04:55.219 --> 00:04:57.219
Another common

00:04:57.219 --> 00:04:59.603
thing people will do with contour plots

00:04:59.603 --> 00:05:01.756
as they represent them is color them.

00:05:01.756 --> 00:05:04.022
So what that might look like

00:05:04.022 --> 00:05:05.636
is here where

00:05:05.636 --> 00:05:09.142
warmer colors like orange
correspond to high values

00:05:09.142 --> 00:05:13.329
and cooler colors like blue
correspond to low values.

00:05:13.329 --> 00:05:15.869
The contour lines end up going along

00:05:15.869 --> 00:05:17.908
the division between red and green here,

00:05:17.908 --> 00:05:20.104
between light green and green,

00:05:20.104 --> 00:05:21.740
and that's another way were

00:05:21.740 --> 00:05:23.855
colors tell you the output and then

00:05:23.855 --> 00:05:25.585
the contour lines
themselves can be thought of

00:05:25.585 --> 00:05:28.755
as the borders between different colors.

00:05:28.755 --> 00:05:30.169
And again a good way to get a feel

00:05:30.169 --> 00:05:31.690
for a multi-dimensional function

00:05:31.690 --> 00:05:34.607
just by looking at the input space.


Fluid flow and vector fields

WEBVTT
Kind: captions
Language: en

00:00:00.940 --> 00:00:01.963
- [Voiceover] So in the last video,

00:00:01.963 --> 00:00:04.553
I talked about Vector fields,
and here I want to talk

00:00:04.553 --> 00:00:07.058
about a special circumstance
where they come up.

00:00:07.058 --> 00:00:10.094
So imagine that we're sitting
in the coordinate plane,

00:00:10.094 --> 00:00:11.572
and that I draw for you a whole bunch

00:00:11.572 --> 00:00:14.885
of little droplets, droplets
of water, and then these are

00:00:14.885 --> 00:00:16.653
going to start flowing in some way.

00:00:17.846 --> 00:00:21.147
How would you describe
this flow mathematically?

00:00:22.095 --> 00:00:24.615
So at every given point, the particles

00:00:24.615 --> 00:00:25.928
are moving in some different way.

00:00:25.928 --> 00:00:28.227
Over here, they're kind of
moving down and to the left.

00:00:28.227 --> 00:00:29.947
Here, they're moving kind of quickly up.

00:00:29.947 --> 00:00:32.016
Over here, they're
moving more slowly down.

00:00:32.016 --> 00:00:34.687
So what you might want
to do is assign a vector

00:00:34.687 --> 00:00:37.642
to every single point in space,

00:00:37.642 --> 00:00:39.911
and a common attribute of
the way that fluids flow.

00:00:39.911 --> 00:00:43.697
This isn't necessarily
obvious, but if you look

00:00:43.697 --> 00:00:46.975
at a given point in space,
let's say like right here,

00:00:46.975 --> 00:00:49.290
every time that a particle
passes through it,

00:00:49.290 --> 00:00:50.972
it's with roughly the same velocity,

00:00:50.972 --> 00:00:53.072
so you might think over time that velocity

00:00:53.072 --> 00:00:54.399
would change, and sometimes it does.

00:00:54.399 --> 00:00:56.241
A lot of times there's some fluid flow

00:00:56.241 --> 00:00:59.530
where it depends on
time, but for many cases

00:00:59.530 --> 00:01:02.737
you can just say, at this point in space,

00:01:02.737 --> 00:01:04.253
whatever particle is going through it,

00:01:04.253 --> 00:01:05.807
it'll have this velocity vector.

00:01:05.807 --> 00:01:08.368
So over here, they might
be pretty high upwards,

00:01:08.368 --> 00:01:11.561
whereas here, it's kind of
a smaller vector downwards,

00:01:11.561 --> 00:01:16.561
even though, I'll play the
animation a little bit more here,

00:01:17.773 --> 00:01:20.651
and if you imagine doing this
at all of the different points

00:01:20.651 --> 00:01:23.444
in space, and assigning
a vector to describe

00:01:23.444 --> 00:01:26.280
the motion of each fluid
particle at each different point,

00:01:26.280 --> 00:01:29.108
what you end up getting is a vector field.

00:01:29.108 --> 00:01:32.216
So this here is a little
bit of a cleaner drawing

00:01:32.216 --> 00:01:35.497
than what I have, and as I
mentioned in the last video,

00:01:35.497 --> 00:01:37.984
it's common for these vectors
not to be drawn to scale,

00:01:37.984 --> 00:01:39.779
but to all have the
same length, just to get

00:01:39.779 --> 00:01:41.764
a sense of direction, and here you can see

00:01:41.764 --> 00:01:45.692
each particle is flowing
roughly along that vector,

00:01:45.692 --> 00:01:47.388
so whatever one it's closest to,

00:01:47.388 --> 00:01:49.439
it's moving in that direction.

00:01:49.439 --> 00:01:52.021
And this is not just a really
good way of understanding

00:01:52.021 --> 00:01:54.297
fluid flow, but it goes
the other way around.

00:01:54.297 --> 00:01:55.691
It's a really good way of understanding

00:01:55.691 --> 00:01:58.531
vector fields themselves,
so sometimes you might

00:01:58.531 --> 00:02:01.245
just be given some new vector field,

00:02:01.245 --> 00:02:03.826
and to get a feel for what it's all about,

00:02:03.826 --> 00:02:06.549
how to interpret it, what
special properties it might have,

00:02:06.549 --> 00:02:08.935
it's actually helpful, even
if it's not meant to represent

00:02:08.935 --> 00:02:11.275
a fluid, to imagine that
it does, and think of all

00:02:11.275 --> 00:02:14.206
the particles, and think of
how they would move along in.

00:02:14.206 --> 00:02:18.977
For example, this particular
one, as you play the animation,

00:02:18.977 --> 00:02:22.754
as you let the particles
move along the vectors,

00:02:22.754 --> 00:02:24.644
there's no change in the density.

00:02:24.644 --> 00:02:26.661
At no point do a punch
of particles go inward,

00:02:26.661 --> 00:02:28.345
or a bunch of particles go outward,

00:02:28.345 --> 00:02:30.305
it stays kind of constant,
and that turns out

00:02:30.305 --> 00:02:33.683
to have a certain mathematical
significance down the road.

00:02:33.683 --> 00:02:35.573
You'll see this later on as we study

00:02:35.573 --> 00:02:37.467
a certain concept called divergence.

00:02:39.105 --> 00:02:40.872
And over here, you see this vector field,

00:02:40.872 --> 00:02:43.267
and you might want to
understand what it's all about,

00:02:43.267 --> 00:02:45.143
and it's kind of helpful
to think of a fluid

00:02:45.143 --> 00:02:48.360
that pushes outward from everywhere,

00:02:48.360 --> 00:02:50.624
and is kind of decreasing in
density around the center,

00:02:50.624 --> 00:02:53.747
and that also has a certain
mathematical significance,

00:02:53.747 --> 00:02:56.665
and it might also lead you to
ask certain other questions.

00:02:56.665 --> 00:02:57.936
Like if you look at the fluid flow

00:02:57.936 --> 00:02:59.735
that we started with in this video,

00:02:59.735 --> 00:03:02.083
you might ask a couple
questions about it like

00:03:02.083 --> 00:03:04.548
it seems to rotating around some points,

00:03:04.548 --> 00:03:06.808
in this case counter
clockwise, but it's rotating

00:03:06.808 --> 00:03:08.383
clockwise around others still.

00:03:09.859 --> 00:03:11.873
Does that have any kind of
mathematical significance?

00:03:12.923 --> 00:03:15.096
Does the fact that there
seem to be the same

00:03:15.096 --> 00:03:16.861
number of particles roughly in this area,

00:03:16.861 --> 00:03:18.518
but they're slowly spilling out there.

00:03:18.518 --> 00:03:20.188
What does that imply for the function

00:03:20.188 --> 00:03:23.540
that represents this whole vector field,

00:03:23.540 --> 00:03:24.705
and you'll see a lot of this later on,

00:03:24.705 --> 00:03:26.929
especially when I talk
about divergence and curl,

00:03:26.929 --> 00:03:29.938
but here I just wanted to
give a little warmup to that

00:03:29.938 --> 00:03:32.931
as we're just visualizing
multivariable functions.


Transformations, part 3

WEBVTT
Kind: captions
Language: en

00:00:00.877 --> 00:00:02.949
- [Voiceover] So I wanna give
you guys just one more example

00:00:02.949 --> 00:00:04.798
of a transformation before we move on

00:00:04.798 --> 00:00:08.048
to the actual calculus of
multi variable calculus.

00:00:08.048 --> 00:00:09.938
In the video on parametric surfaces

00:00:09.938 --> 00:00:12.133
I gave you guys this function here.

00:00:12.133 --> 00:00:14.124
Its a very complicated looking function

00:00:14.124 --> 00:00:15.465
its got a two dimensional input

00:00:15.465 --> 00:00:17.050
and a three dimensional output

00:00:17.050 --> 00:00:18.553
and I talked about how
you can think about it

00:00:18.553 --> 00:00:21.621
as drawing a surface in
three dimensional space

00:00:21.621 --> 00:00:23.185
and that one came out to be the surface

00:00:23.185 --> 00:00:26.944
of a doughnut which we also call a torus.

00:00:26.944 --> 00:00:28.570
So what I wanna do here is talk about

00:00:28.570 --> 00:00:30.662
how you might think of
this as a transformation

00:00:30.662 --> 00:00:32.104
and first let me just get straight

00:00:32.104 --> 00:00:34.380
what the input space here is.

00:00:34.380 --> 00:00:36.331
So the input space you
could think about it

00:00:36.331 --> 00:00:39.434
as the entire T,S plane, right?

00:00:39.434 --> 00:00:42.752
We might draw this as the entire T axis

00:00:44.827 --> 00:00:46.652
and the S axis

00:00:46.652 --> 00:00:49.073
and just everything here
and see where it maps.

00:00:49.073 --> 00:00:51.325
But you can actually go to
just a small subset of that.

00:00:51.325 --> 00:00:54.921
So if you limit yourself
to T going between zero

00:00:54.921 --> 00:00:57.754
so between zero and lets say 2 pi

00:00:59.147 --> 00:01:04.003
and then similarly with S
going from zero up to 2 pi

00:01:04.003 --> 00:01:06.685
can you imagine what, you know

00:01:06.685 --> 00:01:09.570
that would be sort of a square region.

00:01:09.570 --> 00:01:11.419
Just limiting yourself to that

00:01:11.419 --> 00:01:12.963
you're actually gonna
get all of the points

00:01:12.963 --> 00:01:15.686
that you need to draw the torus.

00:01:15.686 --> 00:01:17.230
And the basic reason for that is that

00:01:17.230 --> 00:01:19.384
as T ranges from zero to 2 pi

00:01:19.384 --> 00:01:21.314
the cosine of T goes over its full range

00:01:21.314 --> 00:01:23.935
before it starts becoming periodic.

00:01:23.935 --> 00:01:26.982
Sine of T does the same
and same deal with S.

00:01:26.982 --> 00:01:29.461
If you let S range from zero to 2 pi

00:01:29.461 --> 00:01:30.985
that covers a full period of cosine

00:01:30.985 --> 00:01:32.691
a full period of sine

00:01:32.691 --> 00:01:35.353
so you'll get no new
information by going elsewhere.

00:01:35.353 --> 00:01:36.836
So what we can do is
think about this portion

00:01:36.836 --> 00:01:39.274
of the T,S plane kind of as living inside

00:01:39.274 --> 00:01:41.466
three dimensional space
as a sort of cheating

00:01:41.466 --> 00:01:43.282
but its a little bit easier to do this

00:01:43.282 --> 00:01:44.966
than to imagine moving from

00:01:44.966 --> 00:01:47.462
some separate area into the space.

00:01:47.462 --> 00:01:48.986
At the very least for
the animation efforts

00:01:48.986 --> 00:01:51.524
its easier to just start it off in 3D.

00:01:51.524 --> 00:01:52.704
So what I'm thinking about here

00:01:52.704 --> 00:01:55.813
this square is representing that T,S plane

00:01:55.813 --> 00:01:58.048
and for this function which is taking all

00:01:58.048 --> 00:02:00.120
of the points in this square as its input

00:02:00.120 --> 00:02:02.294
and outputs a point in
three dimensional space

00:02:02.294 --> 00:02:03.879
you can think about how those points

00:02:03.879 --> 00:02:07.379
move to their corresponding output points.

00:02:09.181 --> 00:02:11.437
So I'll show that again.

00:02:11.437 --> 00:02:13.509
We start off with our T,S plane here

00:02:13.509 --> 00:02:15.419
and then whatever your input point is

00:02:15.419 --> 00:02:17.573
if you were to follow it, and you were to

00:02:17.573 --> 00:02:20.559
follow it through this
whole transformation

00:02:20.559 --> 00:02:22.469
the place where it lands would be

00:02:22.469 --> 00:02:25.517
the corresponding output of this function.

00:02:25.517 --> 00:02:26.756
And one thing I should mention is

00:02:26.756 --> 00:02:29.072
all of the interpolating values

00:02:29.072 --> 00:02:31.795
as you go in between
these don't really matter.

00:02:31.795 --> 00:02:33.989
Their function is really
a very static thing

00:02:33.989 --> 00:02:36.021
there's just an input
and there's an output.

00:02:36.021 --> 00:02:37.850
And if I'm thinking in terms of

00:02:37.850 --> 00:02:39.211
a transformation actually moving it

00:02:39.211 --> 00:02:42.238
there's a little bit of magic sauce

00:02:42.238 --> 00:02:45.022
that has to go into making
an animation do this

00:02:45.022 --> 00:02:46.159
and in this case I kind of put it

00:02:46.159 --> 00:02:47.643
into two different phases to sort of

00:02:47.643 --> 00:02:49.878
roll up one side and roll up the other

00:02:49.878 --> 00:02:52.377
it doesn't really matter
but the general idea

00:02:52.377 --> 00:02:54.835
of starting with a square
and somehow warping that

00:02:54.835 --> 00:02:56.745
however you do choose to warp it

00:02:56.745 --> 00:02:58.675
is actually a pretty powerful thought.

00:02:58.675 --> 00:03:00.239
And as we get into multi variable calculus

00:03:00.239 --> 00:03:02.840
and you start thinking a little
more deeply about surfaces

00:03:02.840 --> 00:03:04.587
I think it really helps if you think about

00:03:04.587 --> 00:03:07.391
what a slight little movement over here

00:03:07.391 --> 00:03:10.764
on your input space would look like

00:03:10.764 --> 00:03:12.999
what happens to that tiny little movement

00:03:12.999 --> 00:03:14.563
or that tiny little traversal

00:03:14.563 --> 00:03:16.392
what it looks like if
you do that same movement

00:03:16.392 --> 00:03:18.363
somewhere on the output space.

00:03:18.363 --> 00:03:20.618
And you'll get lots of
chances to wrap your mind

00:03:20.618 --> 00:03:23.198
about this and engage with the idea.

00:03:23.198 --> 00:03:25.128
But here I just want to get
your minds churning on this

00:03:25.128 --> 00:03:28.558
pretty neat way of viewing
what functions are doing.


Multivariable functions

WEBVTT
Kind: captions
Language: en

00:00:00.876 --> 00:00:01.770
- [Voiceover] Hello and welcome

00:00:01.770 --> 00:00:04.034
to multivariable calculus.

00:00:04.034 --> 00:00:05.439
So I think I should probably start off

00:00:05.439 --> 00:00:08.339
by addressing the elephant
in the living room here.

00:00:08.339 --> 00:00:09.816
I am, sadly, not Sal,

00:00:09.816 --> 00:00:11.139
but I'm still gonna teach you some math.

00:00:11.139 --> 00:00:13.618
My name is Grant.

00:00:13.618 --> 00:00:14.912
I'm pretty much a math enthusiast.

00:00:14.912 --> 00:00:17.710
I enjoy making animations
of things when applicable,

00:00:17.710 --> 00:00:18.929
and boy, is that applicable

00:00:18.929 --> 00:00:21.193
when it comes to multivariable calculus.

00:00:21.193 --> 00:00:22.935
So, the first thing we gotta get straight

00:00:22.935 --> 00:00:25.640
is what is this word multivariable

00:00:25.640 --> 00:00:27.172
that separates calculus, as we know it,

00:00:27.172 --> 00:00:30.830
from the new topic that
you're about to study?

00:00:30.830 --> 00:00:33.616
Well, I could say it's all
about multivariable functions,

00:00:33.616 --> 00:00:35.253
that doesn't really answer anything

00:00:35.253 --> 00:00:36.774
because what's a multivariable function?

00:00:36.774 --> 00:00:39.340
And basically, the kinds of functions

00:00:39.340 --> 00:00:40.953
that we're used to dealing with,

00:00:40.953 --> 00:00:44.680
in the old world, in the
ordinary calculus world,

00:00:44.680 --> 00:00:47.373
will have a single input,
some kind of number

00:00:47.373 --> 00:00:48.477
as their input,

00:00:48.477 --> 00:00:50.392
and then the output is
just a single number.

00:00:50.392 --> 00:00:52.575
And you would call this a
single variable function.

00:00:52.575 --> 00:00:56.209
Basically because that guy
there is the single variable.

00:00:56.209 --> 00:00:58.161
So then a multivariable function

00:00:58.161 --> 00:01:01.250
is something that handles
multiple variables.

00:01:01.250 --> 00:01:03.697
So, you know, it's common
to write it as x, y,

00:01:03.697 --> 00:01:05.439
it doesn't really matter
what letters to use,

00:01:05.439 --> 00:01:07.749
and it could be, you know, x, y, z,

00:01:07.749 --> 00:01:09.874
x one, x two, x three,
a whole bunch of things,

00:01:09.874 --> 00:01:11.360
but just to get started,

00:01:11.360 --> 00:01:13.310
we often think just two variables

00:01:13.310 --> 00:01:15.133
and this will output
something that depends on

00:01:15.133 --> 00:01:16.889
both of those.

00:01:16.889 --> 00:01:18.639
Commonly it will output just a number,

00:01:18.639 --> 00:01:20.218
so you might imagine a
number that depends on

00:01:20.218 --> 00:01:23.992
x and y in some way,
like, x squared plus y,

00:01:23.992 --> 00:01:26.151
but it could also output a vector, right?

00:01:26.151 --> 00:01:29.160
So you could also imagine something that's

00:01:29.160 --> 00:01:33.500
got multivariable input, f of x, y,

00:01:33.500 --> 00:01:35.578
and it outputs something that also has

00:01:35.578 --> 00:01:37.552
multiple variables, like,

00:01:37.552 --> 00:01:39.270
I mean I'm just making stuff up here,

00:01:39.270 --> 00:01:42.080
three x and, you know, two y.

00:01:42.080 --> 00:01:44.332
And, this isn't set in stone,

00:01:44.332 --> 00:01:46.039
but the convention is to usually think

00:01:46.039 --> 00:01:48.315
if there's multiple numbers
that go into the output,

00:01:48.315 --> 00:01:49.452
think of it as a vector,

00:01:49.452 --> 00:01:51.438
if there's multiple numbers
that go into the input,

00:01:51.438 --> 00:01:52.621
just kind of write them,

00:01:52.621 --> 00:01:53.951
write them more sideways like this,

00:01:53.951 --> 00:01:56.380
and think of them as a point in space.

00:01:56.380 --> 00:01:58.311
Because, I mean when you
look at something like this,

00:01:58.311 --> 00:02:00.435
and you've got an x and you've got a y,

00:02:00.435 --> 00:02:02.653
you could think about those
as two separate numbers.

00:02:02.653 --> 00:02:03.837
You know, here's your number line

00:02:03.837 --> 00:02:05.730
with the point x on it somewhere,

00:02:05.730 --> 00:02:07.168
maybe that's five, maybe that's three,

00:02:07.168 --> 00:02:08.411
it doesn't really matter.

00:02:08.411 --> 00:02:10.637
And then you've got another number line

00:02:10.637 --> 00:02:12.150
and it's y, and you could think of them

00:02:12.150 --> 00:02:13.520
as separate entities.

00:02:13.520 --> 00:02:15.900
But, it would probably be more accurate

00:02:15.900 --> 00:02:18.396
to call it multidimensional calculus,

00:02:18.396 --> 00:02:20.977
because, really, instead
of thinking of, you know,

00:02:20.977 --> 00:02:22.877
x and y as separate entities,

00:02:22.877 --> 00:02:24.538
whenever you see two things like that

00:02:24.538 --> 00:02:26.569
you're gonna be thinking
about the x y plane.

00:02:26.569 --> 00:02:28.090
And thinking about just a single point.

00:02:28.090 --> 00:02:30.081
And you'd think of this as a
function that takes a point

00:02:30.081 --> 00:02:31.022
to a number,

00:02:31.022 --> 00:02:32.589
or a point to a vector.

00:02:32.589 --> 00:02:33.866
And a lot of people, when they start

00:02:33.866 --> 00:02:35.201
teaching multivariable calculus,

00:02:35.201 --> 00:02:36.606
they just jump into the calculus,

00:02:36.606 --> 00:02:37.546
and there's lots of fun things,

00:02:37.546 --> 00:02:39.268
partial derivatives, gradients,

00:02:39.268 --> 00:02:40.569
good stuff that you'll learn.

00:02:40.569 --> 00:02:41.865
But I think first of all,

00:02:41.865 --> 00:02:43.470
I want to spend a couple videos

00:02:43.470 --> 00:02:44.965
just talking about the different ways

00:02:44.965 --> 00:02:47.065
we visualize the different types

00:02:47.065 --> 00:02:49.215
of multivariable functions.

00:02:49.215 --> 00:02:50.608
So, as a sneak peak,

00:02:50.608 --> 00:02:52.105
I'm just gonna go through a couple of them

00:02:52.105 --> 00:02:53.359
really quickly right now,

00:02:53.359 --> 00:02:55.159
just so you kind of whet your appetite

00:02:55.159 --> 00:02:56.447
and see what I'm getting at,

00:02:56.447 --> 00:02:58.038
but the next few videos are
going to go through them

00:02:58.038 --> 00:03:00.128
in much, much more detail.

00:03:00.128 --> 00:03:03.356
So, first of all, graphs.

00:03:03.356 --> 00:03:04.981
When you have multivariable functions,

00:03:04.981 --> 00:03:07.024
graphs become three dimensional.

00:03:07.024 --> 00:03:08.522
But these only really apply
to functions that have

00:03:08.522 --> 00:03:10.228
some kind of two-dimensional input,

00:03:10.228 --> 00:03:11.436
which you might think about as living

00:03:11.436 --> 00:03:12.667
on this x y plane,

00:03:12.667 --> 00:03:14.180
and a single number as their output

00:03:14.180 --> 00:03:15.697
and the height of the graph

00:03:15.697 --> 00:03:17.589
is gonna correspond with that output.

00:03:17.589 --> 00:03:19.586
Like I said, you'll be
able to learn much more

00:03:19.586 --> 00:03:21.636
about that in the dedicated video on it,

00:03:21.636 --> 00:03:23.986
but these functions also can be visualized

00:03:23.986 --> 00:03:26.599
just in two dimensions,
flattening things out.

00:03:26.599 --> 00:03:28.978
Where we visualize the entire input space

00:03:28.978 --> 00:03:31.079
in associated color, with each point.

00:03:31.079 --> 00:03:33.079
So this is the kind of
thing where you, you know,

00:03:33.079 --> 00:03:34.076
you have some function

00:03:34.076 --> 00:03:36.166
that's got a two-dimensional input,

00:03:36.166 --> 00:03:37.987
that would be f of x, y,

00:03:37.987 --> 00:03:39.764
and what we're looking
at is the x y plane,

00:03:39.764 --> 00:03:41.169
all of the input space,

00:03:41.169 --> 00:03:42.783
and this output's just some number,

00:03:42.783 --> 00:03:44.002
you know, maybe it's like x squared,

00:03:44.002 --> 00:03:45.406
this particular one is an x squared,

00:03:45.406 --> 00:03:46.406
but, you know that,

00:03:46.406 --> 00:03:48.633
and maybe some complicated thing,

00:03:48.633 --> 00:03:50.759
and the color tells you roughly the size

00:03:50.759 --> 00:03:52.698
of that output, and the lines here,

00:03:52.698 --> 00:03:54.207
called contour lines,

00:03:54.207 --> 00:03:58.282
tell you which inputs all
share a constant output value.

00:03:58.282 --> 00:04:00.163
And again, I'll go into
much more detail there.

00:04:00.163 --> 00:04:01.858
These are really nice,
much more convenient

00:04:01.858 --> 00:04:03.117
than three-dimensional graphs,

00:04:03.117 --> 00:04:04.819
to just sketch out.

00:04:04.819 --> 00:04:07.826
Moving right along,

00:04:07.826 --> 00:04:09.683
I'm also gonna talk about surfaces

00:04:09.683 --> 00:04:10.705
in three-dimensional space.

00:04:10.705 --> 00:04:11.819
They look like graphs,

00:04:11.819 --> 00:04:14.125
but they actually deal with
a much different animal,

00:04:14.125 --> 00:04:16.440
that you could think of it
as mapping two dimensions,

00:04:16.440 --> 00:04:18.158
and I like to sort of spoosh it about.

00:04:18.158 --> 00:04:20.504
And we've got kind of a
two-dimensional input,

00:04:20.504 --> 00:04:22.965
that somehow moves into three dimensions,

00:04:22.965 --> 00:04:24.660
and you're just looking at what the output

00:04:24.660 --> 00:04:26.048
of that looks like,

00:04:26.048 --> 00:04:28.264
not really caring about how it gets there.

00:04:28.264 --> 00:04:31.022
These are called parametric surfaces.

00:04:31.022 --> 00:04:33.147
Another fun one is a vector field,

00:04:33.147 --> 00:04:35.504
where every input point is associated

00:04:35.504 --> 00:04:36.642
with some kind of vector,

00:04:36.642 --> 00:04:38.139
which is the output of the function there.

00:04:38.139 --> 00:04:39.068
So this would be a function

00:04:39.068 --> 00:04:40.287
with a two-dimensional input

00:04:40.287 --> 00:04:41.820
and a two-dimensional output

00:04:41.820 --> 00:04:43.724
'cause each of these are
two-dimensional vectors.

00:04:43.724 --> 00:04:44.943
And the fun part with these guys

00:04:44.943 --> 00:04:47.489
is that you can just kind of,

00:04:47.489 --> 00:04:49.099
imagine a fluid flowing,

00:04:49.099 --> 00:04:51.188
so here's a bunch of droplets, like water,

00:04:51.188 --> 00:04:52.779
and they kind of flow along that.

00:04:52.779 --> 00:04:54.300
And that actually turns
out to give insight

00:04:54.300 --> 00:04:55.821
about the underlying function.

00:04:55.821 --> 00:04:57.389
It's one of those beautiful aspects

00:04:57.389 --> 00:04:58.689
of multivariable calc.

00:04:58.689 --> 00:05:00.361
And we'll get lots of exposure to that.

00:05:00.361 --> 00:05:01.603
Again, I'm just sort of zipping through

00:05:01.603 --> 00:05:02.787
to whet your appetite.

00:05:02.787 --> 00:05:06.061
Don't worry if this doesn't
make sense immediately.

00:05:06.061 --> 00:05:07.965
And one of my all-time
favorite ways to think about

00:05:07.965 --> 00:05:11.321
multivariable functions is
to just take the input space,

00:05:11.321 --> 00:05:14.118
in this case, this is gonna
be a function that inputs

00:05:14.118 --> 00:05:15.332
points in two-dimensional space,

00:05:15.332 --> 00:05:17.520
and watch them move to their output,

00:05:17.520 --> 00:05:19.622
so, this is gonna be a
function that also outputs

00:05:19.622 --> 00:05:20.539
in two dimensions.

00:05:20.539 --> 00:05:22.106
And I'm just gonna
watch every single point

00:05:22.106 --> 00:05:24.962
move over to where it's supposed to go.

00:05:24.962 --> 00:05:27.064
These can be kind of
complicated to look at,

00:05:27.064 --> 00:05:28.178
or to think about at first,

00:05:28.178 --> 00:05:30.198
but as you gain a little bit of thought

00:05:30.198 --> 00:05:31.220
and exposure to them,

00:05:31.220 --> 00:05:32.822
they're actually very nice,

00:05:32.822 --> 00:05:34.889
and it provides a beautiful connection

00:05:34.889 --> 00:05:36.224
with linear algebra.

00:05:36.224 --> 00:05:37.280
A lot of you out there,

00:05:37.280 --> 00:05:38.821
if you're studying multivariable calculus,

00:05:38.821 --> 00:05:40.880
you either are about to
study linear algebra,

00:05:40.880 --> 00:05:43.422
or you just have, or maybe
you're doing it concurrently,

00:05:43.422 --> 00:05:45.767
but understanding functions
as transformations

00:05:45.767 --> 00:05:47.846
is gonna be a great way
to connect those two.

00:05:47.846 --> 00:05:48.867
So with that,

00:05:48.867 --> 00:05:51.758
I'll stop jabbering through
these topics really quickly

00:05:51.758 --> 00:05:53.941
and in the next few videos

00:05:53.941 --> 00:05:55.543
I'll actually go through them in detail

00:05:55.543 --> 00:05:57.366
and hopefully you can get a good feel

00:05:57.366 --> 00:06:00.581
for what multivariable functions
can actually feel like.


Transformations, part 2

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.767
- [Voiceover] So in the
last video, I introduced

00:00:01.767 --> 00:00:04.262
transformations and how you
can think about functions

00:00:04.262 --> 00:00:07.107
as moving points in one
space to points in another,

00:00:07.107 --> 00:00:09.823
and here I want to show an
example of what that looks like

00:00:09.823 --> 00:00:12.668
when the input space is two-dimensional.

00:00:12.668 --> 00:00:14.526
So this over here is the input space.

00:00:14.526 --> 00:00:16.325
It's just a copy of the XY plane,

00:00:16.325 --> 00:00:18.925
and the output space is also
two-dimensional, so the output

00:00:18.925 --> 00:00:22.026
space, in this case, also
a two-dimensional plane.

00:00:22.026 --> 00:00:25.079
And what I'm gonna do,
I'm just gonna first play

00:00:25.079 --> 00:00:27.378
an example of one of these transformations

00:00:27.378 --> 00:00:29.920
and then go through the details
of the underyling function

00:00:29.920 --> 00:00:33.067
and how you can understand the
transformation as a result.

00:00:33.067 --> 00:00:35.446
So here's what it looks like.

00:00:35.446 --> 00:00:37.977
Here's what we're gonna be going towards.

00:00:37.977 --> 00:00:40.381
Very complicated, a lot of points moving.

00:00:40.381 --> 00:00:42.204
Lots of different things happening here,

00:00:42.204 --> 00:00:44.468
and what's common with this sort of thing,

00:00:44.468 --> 00:00:46.360
when you're thinking about moving from

00:00:46.360 --> 00:00:48.043
two dimensions to two dimensions,

00:00:48.043 --> 00:00:50.145
given that it's really the
same space, the XY plane,

00:00:50.145 --> 00:00:53.268
you often just think about
the input and output space

00:00:53.268 --> 00:00:55.439
all at once, and instead just watch

00:00:55.439 --> 00:00:58.504
a copy of that plane move onto itself.

00:00:58.504 --> 00:01:00.698
And by the way, when I
say watch, I don't mean

00:01:00.698 --> 00:01:02.637
that you'll always have
an animation like this

00:01:02.637 --> 00:01:04.320
just sort of sitting in front of you.

00:01:04.320 --> 00:01:06.747
When I think about transformations,
it's usually a very

00:01:06.747 --> 00:01:09.185
vague thought in the back
of my mind somewhere,

00:01:09.185 --> 00:01:10.961
but it helps to understand what's

00:01:10.961 --> 00:01:12.540
really going on with the function.

00:01:12.540 --> 00:01:13.957
I'll talk about that more at the end,

00:01:13.957 --> 00:01:16.465
but first let's just go
into what this function is.

00:01:16.465 --> 00:01:19.739
So, the one that I told the
computer to animate here

00:01:19.739 --> 00:01:24.324
is f of x,y is the input, is equal to

00:01:24.324 --> 00:01:27.726
x squared plus y squared,

00:01:27.726 --> 00:01:30.165
as the x component of the output,

00:01:30.165 --> 00:01:32.603
and x squared minus y squared

00:01:32.603 --> 00:01:35.098
is the y component of the output.

00:01:35.098 --> 00:01:37.746
So just to help start understanding this,

00:01:37.746 --> 00:01:40.706
let's take a relatively
simple point like the origin.

00:01:40.706 --> 00:01:42.923
So here, the origin, which is zero, zero,

00:01:42.923 --> 00:01:45.060
and let's think about
what happens to that.

00:01:45.060 --> 00:01:46.999
f of zero, zero.

00:01:46.999 --> 00:01:50.164
Well, x and y are both
zero, so that top is zero.

00:01:51.944 --> 00:01:55.126
And same with the bottom,
that bottom also equals zero.

00:01:55.126 --> 00:01:58.551
Which means, it's taking
the 0.00 to itself,

00:01:58.551 --> 00:02:01.407
and if you watch the
transformation, what this means

00:02:01.407 --> 00:02:05.354
is that the 0.0 stays fixed,
it's like you can hold

00:02:05.354 --> 00:02:07.885
your thumb down on it, and
nothing really happens to it.

00:02:07.885 --> 00:02:10.985
And in fact, we call this a
fixed point of the function

00:02:10.985 --> 00:02:13.516
as a whole, and that kind of terminology

00:02:13.516 --> 00:02:15.316
doesn't really make sense unless

00:02:15.316 --> 00:02:17.673
you're thinking of the
function as a transformation.

00:02:17.673 --> 00:02:19.936
So let's look at another example here.

00:02:19.936 --> 00:02:22.873
Let's take a point like one, one.

00:02:22.873 --> 00:02:25.614
f of one, one.

00:02:25.614 --> 00:02:28.516
So in the input space, let's just kind of

00:02:28.516 --> 00:02:31.790
start this thing over so we're
only looking at the input.

00:02:31.790 --> 00:02:35.169
In the input space, one,
one, is sitting right here,

00:02:35.169 --> 00:02:37.409
and we're wondering
where that's gonna move.

00:02:37.409 --> 00:02:39.952
So when we plug it in,
x squared plus y squared

00:02:39.952 --> 00:02:44.515
is gonna be one squared plus one squared,

00:02:44.515 --> 00:02:46.848
and on the bottom, x
squared minus y squared,

00:02:46.848 --> 00:02:49.693
one squared minus y squared.

00:02:49.693 --> 00:02:51.794
Woop, (laughs) minus one squared.

00:02:51.794 --> 00:02:54.411
I'm plugging things in here.

00:02:54.411 --> 00:02:56.705
So that's two, zero.

00:02:58.565 --> 00:03:00.420
Two, zero.

00:03:02.710 --> 00:03:05.436
Which means we expect this point to move

00:03:05.436 --> 00:03:07.457
over to two, zero in some way.

00:03:07.457 --> 00:03:09.836
So if we watch the
transformation, we expect

00:03:09.836 --> 00:03:11.903
to watch that point move over to here,

00:03:11.903 --> 00:03:13.435
and again, it can be hard to follow

00:03:13.435 --> 00:03:15.014
because there's a lot of moving parts,

00:03:15.014 --> 00:03:17.058
but if you're careful as you watch it,

00:03:17.058 --> 00:03:19.937
the point will actually land right there.

00:03:19.937 --> 00:03:22.735
And you can, in principle,
do this for any given point

00:03:22.735 --> 00:03:24.883
and understand how it
moves from one to another,

00:03:24.883 --> 00:03:27.216
but you might ask, hey Grant,

00:03:27.216 --> 00:03:28.900
what is the point of all of this?

00:03:28.900 --> 00:03:30.780
We have other ways of
visualizing functions that are

00:03:30.780 --> 00:03:33.254
more precise, and kinda less
confusing, to be honest.

00:03:33.254 --> 00:03:35.634
Factor fields are a great
way for functions like this,

00:03:35.634 --> 00:03:37.700
graphs were a great way for functions

00:03:37.700 --> 00:03:39.558
with one input and one output.

00:03:39.558 --> 00:03:42.054
Why think in terms of transformations?

00:03:42.054 --> 00:03:44.143
And the main reason is conceptual.

00:03:44.143 --> 00:03:46.001
It's not like you'll have an
animation sitting in front

00:03:46.001 --> 00:03:48.021
of you, and it's not
like you're gonna by hand

00:03:48.021 --> 00:03:50.819
evaluate a bunch of points
and think of how they move.

00:03:50.819 --> 00:03:53.733
But there's a lot of different
concepts in math, and with

00:03:53.733 --> 00:03:56.102
functions, where when you
understand it in terms of

00:03:56.102 --> 00:03:59.620
a transformation, it gives you
a more nuanced understanding.

00:03:59.620 --> 00:04:02.336
Things like derivatives, or the
variations of the derivative

00:04:02.336 --> 00:04:04.600
that you're gonna learn with
multi-variable calculus,

00:04:04.600 --> 00:04:06.679
there's different ways of understanding it

00:04:06.679 --> 00:04:08.780
in terms of stretching or squishing space

00:04:08.780 --> 00:04:10.695
and things like this, that doesn't really

00:04:10.695 --> 00:04:13.098
have a good analog in terms
of graphs or vector fields.

00:04:13.098 --> 00:04:15.374
So it adds a new color
to your understanding.

00:04:15.374 --> 00:04:17.801
Also, transformations are a super

00:04:17.801 --> 00:04:19.821
important part of linear algebra.

00:04:19.821 --> 00:04:21.493
There will come a point
when you start learning

00:04:21.493 --> 00:04:23.455
the connection between linear algebra

00:04:23.455 --> 00:04:25.359
and multi-variable calculus.

00:04:25.359 --> 00:04:28.737
And if you have a strong
conception of transformations,

00:04:28.737 --> 00:04:30.792
both in the context of linear algebra

00:04:30.792 --> 00:04:32.998
and in the context of
multi-variable calculus,

00:04:32.998 --> 00:04:35.355
you'll be in a much better
position to understand

00:04:35.355 --> 00:04:37.827
the connection between those two fields.


3d vector fields, introduction

WEBVTT
Kind: captions
Language: en

00:00:00.445 --> 00:00:01.385
- [Voiceover] So in the last video,

00:00:01.385 --> 00:00:03.104
I talked about vector
fields in the context

00:00:03.104 --> 00:00:06.064
of two dimensions, and here,
I'd like to do the same

00:00:06.064 --> 00:00:07.666
but for three-dimensions.

00:00:07.666 --> 00:00:09.570
So a three-dimensional vector field

00:00:09.570 --> 00:00:12.902
is given by a function, a
certain multi-variable function

00:00:12.902 --> 00:00:14.690
that has a three-dimensional input

00:00:14.690 --> 00:00:16.769
given with coordinates x, y and z,

00:00:16.769 --> 00:00:19.683
and then a three-dimensional vector output

00:00:19.683 --> 00:00:21.784
that has expressions that
are somehow dependent

00:00:21.784 --> 00:00:23.781
on x, y, and z, I'll just
put dots in here for now,

00:00:23.781 --> 00:00:26.207
but we'll fill this in with
an example in just a moment.

00:00:26.207 --> 00:00:27.426
And the way that this works,

00:00:27.426 --> 00:00:29.110
just like with the
two-dimensinal vector field,

00:00:29.110 --> 00:00:31.246
you're gonna choose a
sample of various points

00:00:31.246 --> 00:00:32.941
in three-dimensional space.

00:00:32.941 --> 00:00:34.411
And for each one of those points,

00:00:34.411 --> 00:00:36.849
you consider what the
output of the function is

00:00:36.849 --> 00:00:38.985
and that's gonna be some
three-dimensional vector.

00:00:38.985 --> 00:00:42.770
And you draw that vector
off of the point itself.

00:00:42.770 --> 00:00:46.572
So to start off, let's
take a very simple example,

00:00:46.572 --> 00:00:50.108
one where the vector that outputs

00:00:50.108 --> 00:00:52.070
is actually just a constant.

00:00:52.070 --> 00:00:54.705
So in this case, I'll make
that constant the vector,

00:00:54.705 --> 00:00:57.585
one, zero, zero.

00:00:57.585 --> 00:01:00.568
So what this vector is,
it's just got a unit lenth

00:01:00.568 --> 00:01:03.482
in the x direction, so this is the x axis.

00:01:03.482 --> 00:01:05.909
So all of the vectors
are gonna end up looking

00:01:05.909 --> 00:01:07.785
something like this where it's a vector

00:01:07.785 --> 00:01:10.722
that has length one in the x direction.

00:01:10.722 --> 00:01:14.182
And when we do this, at
every possible point,

00:01:14.182 --> 00:01:15.378
well not every possible point,

00:01:15.378 --> 00:01:20.006
but a sample of a whole bunch of points,

00:01:20.006 --> 00:01:22.692
we get a vector field
that looks like this.

00:01:22.692 --> 00:01:24.967
At any given point in space,

00:01:24.967 --> 00:01:26.349
we get one of these little blue vectors

00:01:26.349 --> 00:01:27.754
and all of them are the same,

00:01:27.754 --> 00:01:29.170
they're just copies of each other,

00:01:29.170 --> 00:01:33.895
each pointing with unit
length in the x direction.

00:01:33.895 --> 00:01:38.307
So as vector fields go,
this is relatively boring,

00:01:38.307 --> 00:01:39.967
but we can make it a
little bit more exciting

00:01:39.967 --> 00:01:43.392
if we make the input
start to depend somehow

00:01:43.392 --> 00:01:44.588
on the actual input.

00:01:44.588 --> 00:01:48.489
So what I'll do to start,
I'll just make the input

00:01:48.489 --> 00:01:49.952
y, zero, zero.

00:01:49.952 --> 00:01:51.833
So they're still just gonna
point in the x direction,

00:01:51.833 --> 00:01:53.888
but now it's gonna depend on the y value.

00:01:53.888 --> 00:01:56.872
So let's think of a second
before I change the image,

00:01:56.872 --> 00:01:58.207
what that's gonna mean.

00:01:58.207 --> 00:01:59.867
The y axis is this one here,

00:01:59.867 --> 00:02:01.910
so now the z axis is pointing
straight in our face,

00:02:01.910 --> 00:02:03.327
that's the y.

00:02:03.327 --> 00:02:07.228
So as y increases value
to one, two, three,

00:02:07.228 --> 00:02:09.689
the length of these
vectors are gonna increase,

00:02:09.689 --> 00:02:11.547
it's gonna be a stronger
vector in the x direction,

00:02:11.547 --> 00:02:13.590
a very strong vector in the x direction.

00:02:13.590 --> 00:02:16.028
And if y is negative, these
vectors are gonna point

00:02:16.028 --> 00:02:17.456
in the opposite direction.

00:02:17.456 --> 00:02:21.090
So let's see what that looks like.

00:02:21.090 --> 00:02:22.146
Here we go.

00:02:22.146 --> 00:02:24.817
So in this vector field, color and length

00:02:24.817 --> 00:02:28.277
are used to indicate the
magnitude of the vector.

00:02:28.277 --> 00:02:29.890
So red vectors are very long,

00:02:29.890 --> 00:02:30.993
blue vectors are pretty short,

00:02:30.993 --> 00:02:32.189
and at zero, we don't even see any

00:02:32.189 --> 00:02:35.556
because those are
vectors with zero length.

00:02:35.556 --> 00:02:37.588
And just like with two
dimensional vector fields,

00:02:37.588 --> 00:02:39.596
when you draw them, you lie a little bit.

00:02:39.596 --> 00:02:41.895
This one should have a
length of one, right?

00:02:41.895 --> 00:02:43.532
Because when y is equal to one,

00:02:43.532 --> 00:02:44.856
this should have a unit length,

00:02:44.856 --> 00:02:46.307
but it's made really, really small.

00:02:46.307 --> 00:02:49.267
And this one up here,
where y is five or six,

00:02:49.267 --> 00:02:50.951
should be a really long vector,

00:02:50.951 --> 00:02:52.274
but we're lying a little bit

00:02:52.274 --> 00:02:54.248
because if we actually drew them to scale,

00:02:54.248 --> 00:02:56.593
it would really clutter up the image.

00:02:56.593 --> 00:02:58.927
So a couple things to
notice about this one,

00:02:58.927 --> 00:03:02.607
since the output doesn't depend on x or z,

00:03:02.607 --> 00:03:04.175
if you move in the x direction,

00:03:04.175 --> 00:03:05.626
which is back and forth here,

00:03:05.626 --> 00:03:06.868
the vectors don't change.

00:03:06.868 --> 00:03:08.192
And if you move in the z direction

00:03:08.192 --> 00:03:11.013
which is up and down, the
vectors also don't change.

00:03:11.013 --> 00:03:14.275
They only change as you
move in the y direction.

00:03:14.275 --> 00:03:16.957
Okay, so we're starting to
get a feel for how the output

00:03:16.957 --> 00:03:18.432
can depend on the input.

00:03:18.432 --> 00:03:23.273
Now let's do something
a little bit different.

00:03:23.273 --> 00:03:24.736
Let's say that all three of the components

00:03:24.736 --> 00:03:27.789
of the input depend on x, y, and z,

00:03:27.789 --> 00:03:30.587
but I'm just gonna make it
kind of an identity function.

00:03:30.587 --> 00:03:32.329
At a given point x, y, z,

00:03:32.329 --> 00:03:35.336
you output the vector itself, x, y, z.

00:03:35.336 --> 00:03:37.855
So let's think about what
this would actually mean.

00:03:37.855 --> 00:03:39.906
And let's say you've got a given point,

00:03:39.906 --> 00:03:42.553
some point floating off in space.

00:03:42.553 --> 00:03:44.654
What is the output vector for that?

00:03:44.654 --> 00:03:46.442
Well the point has a certain x component,

00:03:46.442 --> 00:03:49.867
a certain y component, and a z component.

00:03:49.867 --> 00:03:52.665
And the vector that corresponds to x, y, z

00:03:52.665 --> 00:03:56.787
is gonna be the one from the
origin to that point itself.

00:03:56.787 --> 00:03:58.726
Let me just draw that here

00:03:58.726 --> 00:04:01.698
from the origin to the point itself.

00:04:01.698 --> 00:04:03.486
And because of how we do vector fields,

00:04:03.486 --> 00:04:06.481
you move this so that instead
of stemming from the origin,

00:04:06.481 --> 00:04:09.418
it actually stems from the point.

00:04:09.418 --> 00:04:11.288
But the main thing to take away from here

00:04:11.288 --> 00:04:14.167
is it's gonna point directly
away from the origin.

00:04:14.167 --> 00:04:15.618
And the farther away the point is,

00:04:15.618 --> 00:04:17.557
the longer this vector will be.

00:04:17.557 --> 00:04:22.312
So with that, let's take a look
at the vector field itself.

00:04:23.559 --> 00:04:25.045
Here we go.

00:04:25.045 --> 00:04:27.727
So again, you kind of
lie when you draw these.

00:04:27.727 --> 00:04:30.247
Like the vectors, these red
guys that are out at the end,

00:04:30.247 --> 00:04:31.466
they should be really long

00:04:31.466 --> 00:04:33.881
'cause this vector should
be as long as that point is

00:04:33.881 --> 00:04:35.599
away from the origin.

00:04:35.599 --> 00:04:37.782
But to give a cleaner vector field,

00:04:37.782 --> 00:04:40.522
you scale things down,
and notice the blue ones

00:04:40.522 --> 00:04:41.779
that are close to the center here,

00:04:41.779 --> 00:04:44.290
are actually really, really short guys.

00:04:44.290 --> 00:04:47.807
And all of these are pointing
directly away from the origin.

00:04:47.807 --> 00:04:48.968
And this is one of those vector fields

00:04:48.968 --> 00:04:50.106
that is actually pretty,

00:04:50.106 --> 00:04:52.660
a good one to have a strong intuition of

00:04:52.660 --> 00:04:53.879
'cause it comes up now and then,

00:04:53.879 --> 00:04:55.720
thinking about what the
identity function looks like

00:04:55.720 --> 00:04:57.993
as a vector field itself.

00:04:57.993 --> 00:05:00.372
In the next video, I'll
talk through another example

00:05:00.372 --> 00:05:02.496
that's a little bit more
complicated than this

00:05:02.496 --> 00:05:04.516
and can hopefully give
an even stronger feel

00:05:04.516 --> 00:05:08.702
for how the output can
depend on x, y, and z.


Interpreting graphs with slices

WEBVTT
Kind: captions
Language: en

00:00:00.554 --> 00:00:01.785
- [Voiceover] So in the last video

00:00:01.785 --> 00:00:04.942
I described how to interpret
three dimensional graphs

00:00:04.942 --> 00:00:07.142
and I have another three
dimensional graph here,

00:00:07.142 --> 00:00:08.804
it's a very bumpy guy.

00:00:08.804 --> 00:00:12.971
And this happens to be the graph
with the function f of x, y

00:00:14.879 --> 00:00:16.796
is equal to cosine of x

00:00:19.419 --> 00:00:21.669
multiplied by the sine of y

00:00:26.110 --> 00:00:27.610
and you know I could also say like,

00:00:27.610 --> 00:00:30.430
that this graph represents

00:00:30.430 --> 00:00:32.101
I could also say that
this graph represents

00:00:32.101 --> 00:00:34.820
z is equal to that whole value

00:00:34.820 --> 00:00:37.300
because we think about
the output of the function

00:00:37.300 --> 00:00:39.522
as the z coordinate of each point

00:00:39.522 --> 00:00:41.498
and what I want to do here is describe

00:00:41.498 --> 00:00:44.447
how you can interpret the
relationship between this graph

00:00:44.447 --> 00:00:47.197
and these functions that you know

00:00:48.093 --> 00:00:50.055
by taking slices of it.

00:00:50.055 --> 00:00:53.657
So for example let's say
that we took a slice with

00:00:53.657 --> 00:00:55.326
this plane here

00:00:55.326 --> 00:00:57.552
and what this plane here is

00:00:57.552 --> 00:01:01.135
is it represents the value x equals zero

00:01:04.208 --> 00:01:06.701
and you can kinda see that
because this is the x-axis.

00:01:06.701 --> 00:01:10.047
So when you're at zero on the x-axis

00:01:10.047 --> 00:01:11.431
you know, you pass through the origin

00:01:11.431 --> 00:01:13.932
and then the values of
y and z can go freely

00:01:13.932 --> 00:01:15.432
so you end up with

00:01:16.817 --> 00:01:17.918
this plane.

00:01:17.918 --> 00:01:19.699
And let's say you want to just consider

00:01:19.699 --> 00:01:22.590
where this cuts through the graph, okay,

00:01:22.590 --> 00:01:23.971
so we'll limit our graph

00:01:23.971 --> 00:01:25.588
just down to the point where it cuts it

00:01:25.588 --> 00:01:28.815
and I'm going to draw a little
red line over that spot.

00:01:28.815 --> 00:01:30.162
Now what you might notice here

00:01:30.162 --> 00:01:33.030
that red line looks like a sinusoidal wave

00:01:33.030 --> 00:01:35.955
in fact it looks exactly
like the sine function itself

00:01:35.955 --> 00:01:37.421
you know, passes through the origin

00:01:37.421 --> 00:01:40.758
it starts by going up and this makes sense

00:01:40.758 --> 00:01:42.709
if we start to plug things into

00:01:42.709 --> 00:01:45.452
the original form here.

00:01:45.452 --> 00:01:47.035
Because if you take

00:01:49.410 --> 00:01:51.864
f and you plug in x equals zero

00:01:51.864 --> 00:01:54.846
but then we still let y range freely

00:01:54.846 --> 00:01:59.492
what it means is that you're
looking at cosine of zero

00:01:59.492 --> 00:02:01.409
multiplied by sine of y

00:02:06.181 --> 00:02:08.014
And what is cosine of zero?

00:02:08.014 --> 00:02:11.015
cosine of zero evaluates to one

00:02:11.015 --> 00:02:13.995
so this whole function should
look just like sine of y

00:02:13.995 --> 00:02:17.389
in that when we let y run freely

00:02:17.389 --> 00:02:21.119
the output, which is still
represented by the z coordinate,

00:02:21.119 --> 00:02:22.920
will give us this graph that's just a

00:02:22.920 --> 00:02:24.597
normal two-dimensional graph

00:02:24.597 --> 00:02:26.679
that we're probably familiar with.

00:02:26.679 --> 00:02:28.319
And let's try this at a different point.

00:02:28.319 --> 00:02:30.209
Let's see what would happen if

00:02:30.209 --> 00:02:33.091
instead of plugging in x equals zero

00:02:33.091 --> 00:02:36.924
let's imagine that we
plugged in y equals zero

00:02:38.271 --> 00:02:40.524
and this time before I graph it

00:02:40.524 --> 00:02:43.670
and before I show everything that goes on

00:02:43.670 --> 00:02:45.769
let's just try to figure out

00:02:45.769 --> 00:02:47.734
purely from the formula here

00:02:47.734 --> 00:02:50.789
what it's going to look like
when we plug in y equals zero.

00:02:50.789 --> 00:02:53.013
So now I'm going to write
over on the other side.

00:02:53.013 --> 00:02:54.245
We have

00:02:54.245 --> 00:02:56.862
f of x will still run freely

00:02:56.862 --> 00:02:59.183
y is going to be fixed as zero

00:02:59.183 --> 00:03:01.610
and what this means is we have cosine of x

00:03:01.610 --> 00:03:02.881
so maybe expect to see something

00:03:02.881 --> 00:03:05.368
that looks kinda like a cosine graph

00:03:05.368 --> 00:03:07.201
and then sine of zero.

00:03:08.847 --> 00:03:11.541
Except, what is sine of zero?

00:03:11.541 --> 00:03:14.980
Sine of zero cancels out
and just becomes zero

00:03:14.980 --> 00:03:17.567
which multiplied by cosine of x

00:03:17.567 --> 00:03:20.059
means everything cancels
out and becomes zero

00:03:20.059 --> 00:03:21.309
so what you'd expect is that this

00:03:21.309 --> 00:03:23.127
is going to look like a constant function

00:03:23.127 --> 00:03:25.031
that's constantly equal to zero.

00:03:25.031 --> 00:03:26.633
And let's see if that's what we get.

00:03:26.633 --> 00:03:28.363
So I'm going to slice it with

00:03:28.363 --> 00:03:29.863
y equals zero here

00:03:31.045 --> 00:03:33.054
and you look at the y axis

00:03:33.054 --> 00:03:34.224
we see when it's zero

00:03:34.224 --> 00:03:35.970
and x and z both run freely.

00:03:35.970 --> 00:03:40.042
I'm going to chop off
my graph at that point,

00:03:40.042 --> 00:03:42.830
and indeed it chops it
just at this straight line,

00:03:42.830 --> 00:03:46.997
the straight line that goes
right along the x-axis.

00:03:49.739 --> 00:03:52.292
But let's say that we did a
different constant value of y.

00:03:52.292 --> 00:03:54.459
Rather than y equals zero,

00:03:57.503 --> 00:04:00.573
and we'll erase all of this,

00:04:00.573 --> 00:04:02.823
let's say that I cut things

00:04:04.948 --> 00:04:06.794
at some other value.

00:04:06.794 --> 00:04:09.461
So in this case what I've chosen

00:04:10.323 --> 00:04:12.490
is y is equal to pi halves

00:04:15.467 --> 00:04:16.467
and it looks

00:04:17.811 --> 00:04:20.819
kinda like we've got a wave here

00:04:20.819 --> 00:04:22.281
and it looks like a cosine wave

00:04:22.281 --> 00:04:24.800
and you can probably
see where this is going.

00:04:24.800 --> 00:04:26.651
This is when x is running freely

00:04:26.651 --> 00:04:29.066
and if we start to
imagine plugging this in

00:04:29.066 --> 00:04:32.329
I'll just actually write it out.

00:04:32.329 --> 00:04:34.593
We've got cosine of x

00:04:34.593 --> 00:04:38.760
and then y is held at a
constant sine of pi halves.

00:04:39.832 --> 00:04:42.499
Sine of pi halves is just,

00:04:42.499 --> 00:04:44.241
this just always equals one

00:04:44.241 --> 00:04:45.645
so we could replace this with one

00:04:45.645 --> 00:04:46.876
which means the function as a whole

00:04:46.876 --> 00:04:48.710
should look like cosine x.

00:04:48.710 --> 00:04:49.989
So again the multi-variable function

00:04:49.989 --> 00:04:52.843
we've frozen y and we're
letting x range freely

00:04:52.843 --> 00:04:55.593
and it ends up looking
like a cosine function

00:04:55.593 --> 00:04:58.356
and I think a really
good way to understand

00:04:58.356 --> 00:05:01.395
a given three-dimensional
graph when you see it,

00:05:01.395 --> 00:05:02.562
let's say you,

00:05:04.791 --> 00:05:06.092
you look back at the original graph

00:05:06.092 --> 00:05:08.170
when we don't have anything going on.

00:05:08.170 --> 00:05:09.306
Get rid of that little line.

00:05:09.306 --> 00:05:10.246
So you've got this graph

00:05:10.246 --> 00:05:11.583
and it looks wavy and bumpy

00:05:11.583 --> 00:05:13.534
and a little bit hard
to understand at first

00:05:13.534 --> 00:05:15.269
but if you just think in terms of

00:05:15.269 --> 00:05:17.063
holding one variable constant

00:05:17.063 --> 00:05:19.725
it boils down always into

00:05:19.725 --> 00:05:21.315
a normal two-dimensional graph

00:05:21.315 --> 00:05:22.450
and you can even think about,

00:05:22.450 --> 00:05:24.911
as you're lighting planes,
kinda slide back and forth

00:05:24.911 --> 00:05:26.119
what that means for the

00:05:26.119 --> 00:05:29.417
amplitude of the wave that you see

00:05:29.417 --> 00:05:30.734
and things like that.

00:05:30.734 --> 00:05:32.701
This become especially
important, by the way,

00:05:32.701 --> 00:05:36.868
when we introduce a notion
of partial derivatives.