Partial derivatives  Multivariable Calculus  Khan Academy  Summary and Q&A
TL;DR
This video explains how to visualize functions in three dimensions and how to find the slope of a surface in a specific direction.
Questions & Answers
Q: What are the different types of representations of functions in three dimensions?
Functions in three dimensions can be represented as lines, curves, surfaces, or vector fields. Each type has its own characteristics and can be visualized in different ways.
Q: How do we define a surface in three dimensions?
A surface in three dimensions can be defined by expressing z as a function of x and y. This means that for any given x and y values, we can determine the corresponding z value for the surface.
Q: Why do we use the righthand rule to determine the orientation of the coordinate system?
The righthand rule is used to establish a convention for the orientation of the coordinate system. In this convention, the cross product of the xaxis with the yaxis is equal to the zaxis. It helps determine the direction of increasing x and establishes consistency in the coordinate system.
Q: How do we find the slope of a surface in three dimensions?
To find the slope of a surface in three dimensions, we need to take the partial derivative. This involves holding one variable constant and taking the derivative with respect to the other variable. The result is the slope of the surface in a particular direction.
Summary & Key Takeaways

Functions in three dimensions can take the form of lines, curves, surfaces, or vector fields.

A surface in three dimensions is the most intuitive way to visualize functions.

To plot points on a surface, we can define z as a function of x and y.