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### Course: Multivariable calculus>Unit 1

Lesson 3: Visualizing scalar-valued functions

# Introduction to 3d graphs

Three-dimensional graphs are a way to represent multi-variable functions with two inputs and one output. They're visualized by plotting input-output pairs in 3D space, resulting in a surface. Other methods of visualizing multi-variable functions include contour maps, parametric functions, and vector fields. Created by Grant Sanderson.

## Want to join the conversation?

• who is Grant and does he do more instructional videos?
• Hello, This Very Cool Guy is Grant Sanderson, he has a YouTube Channel named 3blue1brown (I'd like to know what does that mean), where he talks about many different topics of his own choice, these days He's doing series on The essence of Linear Algebra, I recommend that you check them out, video-by-video.
That's all ! Have A Nice Day !
• How difficult is it to visualize more than 3 dimensions? Like, how many people can actually do that? Is it a requirement for very advanced math?
• That's difficult indeed! It would be hard to go beyond 4 dimensions, which could be a 3d object changing in time.
He mentions near the end of the video its tempting to think of all functions in terms of graphs, but difficult to do beyond 3 inputs+outputs. That's why we'll take a look at the 'work around' ways of visualizing those, like contour maps and vector space. Good luck!
• What software is being utilized to generate the graphs? Thank you.
• It's called Grapher. If you have a newer Mac, go to your home screen, and press "command + spacebar", and type in Grapher.
• Are all outputs of multivariable equations 1-dimensional? I am wondering because I noticed that for f(x)=x^2, the input was 1-D and the output was 1-D. For f(x,y)=x^2+y^2, the input was 2-D and the output was 1-D again. So, are all outputs of multivariable functions the "height" of the graph? What would be examples of multidimensional outputs? Thanks!
• There are definitely outputs of more than 1D. Parametric equations such as x = 2t, y = t^2 are an example. The calculus of functions with one input and multiple outputs is pretty trivial (just do the operation on each dimension of output). But when there are multiple inputs and outputs, you need vector calculus (which is taught on Khan Academy).
• As far as graphing utilities, do we only go up to 3D graphs? Is it even possible to have computers map out more than 3D?
• The 4th dimension is cosidered to be time, the different positions of the 3rd dimension at different stages. So if you think about it, an animated movie (or any computer animation) is in 4D.
• Why aren't there any exercises to go with all the videos?
• How can I turn on the subtitle? I want the transcript showing at the bottom of video.
• If you click the link in the lower right corner just below the video that reads "Switch to full player," you will get the version that is hosted on YouTube and you can toggle subtitles on.
• If you multiplied f(x,y) by a number that was small enough, say 1/n as n->infinity, would the output approach the original input plane as it became 'flatter' ?
• It will which means that every output will approach to zero value.
As n -> infinity, 1/n -> zero. This means you are multiplying by more and more zeroish number.
Precisely, lim_n->infinity {(1/n)f(x)}=0
(1 vote)
• Why do the squares in the 3D graph get deformed(some seem elongated,some compressed)in different parts of the graph?
What does it actually signify?
(1 vote)
• The squares are meant to give you an idea of how the graph is curving through space. The deformities you mention help your eye interpret what it is seeing into a "3D" shape. The squares also probably (although I am not sure) represent sections of input, where the vertices of the squares represent integer x,y coordinates.