$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 12: Functions of Several Variables

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

A function of the form $$y=f(x)$$ is a function of a single variable; given a value of $$x$$, we can find a value $$y$$. Even the vector--valued functions of Chapter 11 are single--variable functions; the input is a single variable though the output is a vector. There are many situations where a desired quantity is a function of two or more variables. For instance, wind chill is measured by knowing the temperature and wind speed; the volume of a gas can be computed knowing the pressure and temperature of the gas; to compute a baseball player's batting average, one needs to know the number of hits and the number of at--bats. This chapter studies multivariable functions, that is, functions with more than one input.

• 12.1: Introduction to Multivariable Functions
The graph of a function f of two variables is the set of all points (x,y,f(x,y)) where (x,y) is in the domain of f . This creates a surface in space.
• 12.2: Limits and Continuity of Multivariable Functions
We continue with the pattern we have established in this text: after defining a new kind of function, we apply calculus ideas to it. The previous section defined functions of two and three variables; this section investigates what it means for these functions to be "continuous.''
• 12.3: Partial Derivatives
A partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry.
• 12.4: Differentiability and the Total Differential
We extend this idea to functions of two variables.
• 12.5: The Multivariable Chain Rule
In this section we extend the Chain Rule to functions of more than one variable.
• 12.6: Directional Derivatives
Partial derivatives give us an understanding of how a surface changes when we move in the x and y directions.  But what if we didn't move exactly in x or y directions?  Partial derivatives alone cannot measure this. This section investigates directional derivatives, which do measure this rate of change.
• 12.7: Tangent Lines, Normal Lines, and Tangent Planes
Derivatives and tangent lines go hand-in-hand. When dealing with functions of two variables, the graph is no longer a curve but a surface. At a given point on the surface, it seems there are many lines that fit our intuition of being "tangent'' to the surface.
• 12.8: Extreme Values
Given a function z=f(x,y) , we are often interested in points where z takes on the largest or smallest values.
• 12.E: Applications of Functions of Several Variables (Exercises)