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## Visualizing scalar-valued functions

Current time:0:00Total duration:5:36

# Contour plots

## Video transcript

- [Voiceover] So I have here
a three-dimensional graph, and that means that it's representing some kind of function that has a two-dimensional input
and a one-dimension output. So that might look
something like f of x, y equals and then just some expression that has a
bunch of x's and y's in it. And graphs are great but they're kinda clunky to draw. I mean, certainly you can't
just scribble it down. It typically requires some
kind of graphing software and when you take a static image of it it's not always clear what's going on. So here I'm going to
describe a way that you can represent these functions and
these graphs two-dimensionally just by scribbling down on a
two-dimensional piece of paper. This is a very common way
that you'll see if you're reading a textbook or if someone
is drawing on a blackboard. It's known as a contour plot and the idea of a contour plot is that we're going to take this graph and slice it a bunch of times. So I'm going to slice
it with various planes that are all parallel to the x, y plane and let's think for a moment about what these guys represent. So the bottom one here represents the value z
is equal to negative two. This is the z-axis over here and when we fix that to be negative two and let x and y run freely
we get this whole plane. And if you let z increase,
keep it constant, but let it increase by one to negative one we get a new plane, still
parallel to the x, y plane but it's distance from the
x, y plane is negative one. And the rest of these guys, they're all still constant values of z. Now in terms of our graph, what that means is that these represent constant
values of the graph itself. These represent constant
values for the function itself. So because we always represent
the output of the function as the height off of the x, y plane these represent constant
values for the output. What that's going to look like. So what we do is we say, "Where do these
slices cut into the graph?" So I'm going to draw on all of the points where those slices cut into the graph and these are called contour lines. We're still in three-dimensions so we're not done yet. So what I'm going to do is
take all these contour lines and I'm going to squish them
down on to the x, y plane. So what that means, each of them has some kind
of z component at the moment, and we're just going to chop it down, squish them all nice and flat, on to the x, y plane. And now we have something two-dimensional, and it still represents some of the outputs of our function. Not all of them, it's not perfect, but it does give a very good idea. I'm going to switch over to
a two-dimensional graph here. And this is that same function that we were just looking at. Let's actually move it a
little bit more central here. So this is the same function that we were just looking at, but each of these lines represents a constant output of the function so it's important to realize we're still representing a function that has a
two-dimensional input and a one-dimensional output. It's just that we're
looking in the input space of that function as a whole. So this is still f of x, y and then some expression of those guys but this line might represent
the constant value of f when all of the values
were at outputs three. Over here, this also, both of these circles together give you all the values where f outputs three. This one over here will tell you where it outputs two and you can't know this just
looking at the contour plot so typically if someone's drawing it if it matters that you
know the specific values they'll mark it somehow. They'll let you know what
value each line corresponds to but as soon as you know that
this line corresponds to zero it tells you that every
possible input point that sits somewhere on this line will evaluate to zero when you
pump it through the function. And this actually gives a very good feel for the shape of things. If you like thinking in terms of graphs you can kind of imagine how these circles and everything
would pop out of the page. You can also look, notice how the lines are really
close together over here, very, very close together, but they're a little
more spaced over here. How do you interpret that? Well, over here this means it
takes a very, very small step to increase the value
of the function by one, very small step and it increases by one, but over here it takes a much larger step to increase the function
by the same value. So over here this kind of means steepness. If you see a very short
distance between contour lines it's going to be very steep but over here it's much more shallow. And you can do things like this
to kind of get a better feel for the function as whole. The idea of a whole bunch
of concentric circles usually corresponds to
a maximum or a minimum, and you end up seeing these a lot. Another common thing people will do with contour plots as they represent them is color them. So what that might look like is here where warmer colors like orange
correspond to high values and cooler colors like blue
correspond to low values. The contour lines end up going along the division between red and green here, between light green and green, and that's another way were colors tell you the output and then the contour lines
themselves can be thought of as the borders between different colors. And again a good way to get a feel for a multi-dimensional function just by looking at the input space.