Introduction to 3d graphs  Multivariable calculus  Khan Academy  Summary and Q&A
TL;DR
Threedimensional graphs are a way to represent multivariable functions using two inputs and one output.
Key Insights
 ð Multivariable functions can be represented using threedimensional graphs, which involve two inputs and one output.
 ðĪŠ Inputs in threedimensional graphs are pairs of points, and the natural way to visualize them is by using a triplet of x, y, and z coordinates.
 ð The shape of the graph can be modified by altering the function, such as changing the height of the output for each input.
 â Threedimensional graphs are limited to representing functions with a threedimensional input and onedimensional output, as visualizing higher dimensions becomes challenging.
Transcript
 [Voiceover] Hello everyone. So what I'd like to do here is to describe how we think about threedimensional graphs. Threedimensional graphs are a way that we represent certain kind of multivariable function that kind of has two inputs, or rather a twodimensional input, and then onedimensional of output of some kind. So the one that I have pic... Read More
Questions & Answers
Q: How are twodimensional graphs different from threedimensional graphs?
Twodimensional graphs represent the relationship between a single input and its corresponding output, while threedimensional graphs show the relationship between two inputs and the resulting output.
Q: How are inputoutput pairs visualized in twodimensional graphs?
In twodimensional graphs, inputoutput pairs are plotted as points on the graph. Each input corresponds to an xcoordinate, and the output is represented by the height of the graph above that point.
Q: How are inputoutput pairs visualized in threedimensional graphs?
In threedimensional graphs, inputoutput pairs are represented as points in threedimensional space. The x and y coordinates determine the input, and the z coordinate represents the output.
Q: Is it possible to visualize all possible inputoutput pairs in a threedimensional graph?
No, it is not feasible to plot all possible inputoutput pairs in a threedimensional graph, as there are infinitely many combinations. Instead, a surface is drawn that represents the relationship between the inputs and outputs.
Summary & Key Takeaways

Threedimensional graphs are a visual representation of multivariable functions, showing the relationship between two inputs and one output.

In twodimensional graphs, the inputs are plotted on the xaxis, the outputs are represented by the height of the graph above each point.

In threedimensional graphs, inputs are pairs of points, and the natural way to visualize them is by using a triplet of x, y, and z coordinates.