If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## 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? •  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
• At in the video we see that the contour lines for z=1, not intersecting at point (0,0) in the X-Y plane.

But at in 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? • 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? • Does anyone know what the function is being graphed?
f(x,y) = ....?

Thanks • 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? • 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? • 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.
• At , is that Desmos Graphing Calculator? • 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 • 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.  