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## Multivariable calculus

### Course: Multivariable calculus > Unit 1

Lesson 3: Visualizing scalar-valued functions# Contour plots

An alternative method to representing multivariable functions with a two-dimensional input and a one-dimensional output, contour maps involve drawing purely in the input space. Created by Grant Sanderson.

## Want to join the conversation?

- You are brilliant!Keep up the good work......Is this topic of maths that explains atomic orbitals shapes?(48 votes)
- Orbital shapes are only aproximated idealizations of statistical information about where is more probably that an electron is located at certain moment (this also is an aproximation since Heisenberg found that it is incompatible to describe the position and the time of an electron). Even in a perfect situation where an only atom is isolated from universe gravitation, electromagnetism..., the shape of the orbit (all space points where an electron could be al some point) wouldn't be an sphere, neither a surface. We could only say, for example, the probability of the orbit of this electron is between these two spherical surfaces (now, give two radios from nucleous) is 95%, or 99%, or 99,9999%... but it is imposible to draw an exact surface (remember a surface doesn't have thickness) outside of where an electron would never be (if we also idealize it like a spheric particle, and not a wave or another concept).

P.S.: sorry if my english has many mistakes, I am from Spain(39 votes)

- At2:41in the video we see that the contour lines for z=1, not intersecting at point (0,0) in the X-Y plane.

But at3:46in the video, Grant marks a contour line as z=1 which intersects at pint (0,0) in the X-Y plane.

Can please someone explain this difference?(12 votes)- we're looking at countour plot in x-y plane from ABOVE, not from the front/in the direction your eyes usually look things, so imagine you're flying and seeing a building just right below your feet(4 votes)

- Is there a way to determine if the graph is coming up out of the page or going into it based on the contour plot?(9 votes)
- From what I understand by above video, colour gradient of the contour map can indicate whether graph is coming out or going down.(9 votes)

- Does anyone know what the function is being graphed?

f(x,y) = ....?

Thanks(7 votes)- From my visual examination, the graph that's shown in the video appears to be the same graph represented in this KA article, and the function is given:

https://www.khanacademy.org/math/multivariable-calculus/thinking-about-multivariable-function/ways-to-represent-multivariable-functions/a/contour-maps

The function in question seems to be f(x,y) = x^4 - x^2 + y^2. Hope that helps!(10 votes)

- When I was in middle school, I took a geography course that discussed contour plots as a 2-D method of representing altitude (although of course since it was a middle school course it didn't examine the math involved). I had often wondered how those maps were generated, but no one was able to explain the concept to me. Now I finally begin to understand how that works!

Now I'm curious whether it's possible to develop multivariate equations to model and describe actual surfaces on the earth or on other celestial bodies. So in other words, just like how we might use the Lotka-Volterra Model differential equations to describe an actual predator-prey population relationship, is it similarly possible to find multivariate equations/models that can mathematically represent an actual surface on the earth? If so, how is that done, and can anyone show/describe to me some specific examples of when and how this was done?(8 votes) - Is the ever a need to slice the graphs in a diagonal sense, but keeping the center point of the plane in the z-axis constant?(4 votes)
- Hmmm, nothing comes to mind immediately. You think about diagonal slicing in the context of the directional derivative (a few videos down the road from this one), but in that case the plane you are using to slice things is oriented vertically (i.e., parallel to the z-axis), and it is only "diagonal" in that it intersects the xy-plane along a diagonal line.(7 votes)

- At2:42, is that Desmos Graphing Calculator?(4 votes)
- So we dont need to know what function represents this graph?

Because as i am watching i dont feel comfortable with graph because it is your approach to shape. What can be unique approach to any graph? If there is another complex 3d graph how i can represent it?

Thanks(3 votes)- The contour lines we use to make a contour plot are a set of all x and y values which, together, produce a specific z-value.

If you're working with some other 3D graph then, you'll want to check to find which values of x and y together produce z. The easiest way to do this is to set a fixed value for one variable and then solve for the other. So, if you have a function F (x,y) = 2x + 3y, and you want to create a contour line for z = 3. You'd pick some particular x value, like x=1, and then solve for y (which will give you y = 1/3, but for more complicated functions there could be many more possible y values). Then, repeat this for different x-values until you have an idea of the contour line for that z value. Making a contour plot by hand is just doing this over and over until you have enough points to give you an understanding of what is happening.(4 votes)

- Which software has been used in this vid?(4 votes)
- What does an intersection of contour lines mean? In the case of this graph, it happens on the origin.

Similarly, what do contour curves that are lines represent? How does the slope or derivative of the contour curves relate to the original graph?(2 votes)

## 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.