3.1 Functions of Several Variables

$\newenvironment {prompt}{}{} \newcommand {\ungraded }[0]{} \newcommand {\ffrac }[2]{\frac {\text {\footnotesize $#1$}}{\text {\footnotesize $#2$}}} \newcommand {\npnoround }[0]{\nprounddigits {-1}} \newcommand {\npnoroundexp }[0]{\nproundexpdigits {-1}} \newcommand {\npunitcommand }[1]{\ensuremath {\mathrm {#1}}} \newcommand {\RR }[0]{\mathbb R} \newcommand {\R }[0]{\mathbb R} \newcommand {\C }[0]{\mathbb C} \newcommand {\N }[0]{\mathbb N} \newcommand {\Z }[0]{\mathbb Z} \newcommand {\dis }[0]{\displaystyle } \newcommand {\d }[0]{\mathop {}\!d} \newcommand {\l }[0]{\ell } \newcommand {\ddx }[0]{\frac {d}{\d x}} \newcommand {\ppx }[0]{\frac {\partial }{\partial x}} \newcommand {\ppy }[0]{\frac {\partial }{\partial y}} \newcommand {\zeroOverZero }[0]{\ensuremath {\boldsymbol {\tfrac {0}{0}}}} \newcommand {\inftyOverInfty }[0]{\ensuremath {\boldsymbol {\tfrac {\infty }{\infty }}}} \newcommand {\zeroOverInfty }[0]{\ensuremath {\boldsymbol {\tfrac {0}{\infty }}}} \newcommand {\zeroTimesInfty }[0]{\ensuremath {\small \boldsymbol {0\cdot \infty }}} \newcommand {\inftyMinusInfty }[0]{\ensuremath {\small \boldsymbol {\infty -\infty }}} \newcommand {\oneToInfty }[0]{\ensuremath {\boldsymbol {1^\infty }}} \newcommand {\zeroToZero }[0]{\ensuremath {\boldsymbol {0^0}}} \newcommand {\inftyToZero }[0]{\ensuremath {\boldsymbol {\infty ^0}}} \newcommand {\numOverZero }[0]{\ensuremath {\boldsymbol {\tfrac {\#}{0}}}} \newcommand {\dfn }[0]{\textbf } \newcommand {\unit }[0]{\mathop {}\!\mathrm } \newcommand {\eval }[1]{ #1 \bigg |} \newcommand {\seq }[1]{\left ( #1 \right )} \newcommand {\epsilon }[0]{\varepsilon } \newcommand {\iff }[0]{\Leftrightarrow } \DeclareMathOperator {\arccot }{arccot} \DeclareMathOperator {\arcsec }{arcsec} \DeclareMathOperator {\arccsc }{arccsc} \DeclareMathOperator {\si }{Si} \DeclareMathOperator {\proj }{proj} \DeclareMathOperator {\scal }{scal} \DeclareMathOperator {\cis }{cis} \DeclareMathOperator {\Arg }{Arg} \DeclareMathOperator {\Rep }{Re} \DeclareMathOperator {\Imp }{Im} \DeclareMathOperator {\sech }{sech} \DeclareMathOperator {\csch }{csch} \DeclareMathOperator {\Log }{Log} \newcommand {\tightoverset }[2]{\mathop {#2}\limits ^{\vbox to -.5ex{\kern -0.75ex\hbox {$#1$}\vss }}} \newcommand {\arrowvec }[0]{\overrightarrow } \newcommand {\vec }[0]{\mathbf } \newcommand {\veci }[0]{{\boldsymbol {\hat {\imath }}}} \newcommand {\vecj }[0]{{\boldsymbol {\hat {\jmath }}}} \newcommand {\veck }[0]{{\boldsymbol {\hat {k}}}} \newcommand {\vecl }[0]{\boldsymbol {\l }} \newcommand {\utan }[0]{\vec {\hat {t}}} \newcommand {\unormal }[0]{\vec {\hat {n}}} \newcommand {\ubinormal }[0]{\vec {\hat {b}}} \newcommand {\dotp }[0]{\bullet } \newcommand {\cross }[0]{\boldsymbol \times } \newcommand {\grad }[0]{\boldsymbol \nabla } \newcommand {\divergence }[0]{\grad \dotp } \newcommand {\curl }[0]{\grad \cross } \newcommand {\vector }[1]{\left \langle #1\right \rangle }$

In this section we describe functions of two or more variables.

A function of two variables has the form $z = f(x, y)$ The independent variables are $x$ and $y$ and the dependent variable is $z$ . In section 1.8, we saw elliptic paraboloids like $z = \frac {x^2}{4} + \frac {y^2}{9}$ In this equation, we can view $z$ as a function of $x$ and $y$ .
The area of a rectangle is length times width, so we can express area as a function of two variables: $A = f(l, w) = lw$ The price of a good is a function of supply and demand, so we may be able to write $p = f(s, d)$ The depth of the ocean is a function of the location on its surface, leading to $depth = f(latitude, longitude)$ The domain of $f(x, y)$ is the set of coordinate pairs $(x, y)$ in $\R ^2$ for which $f(x, y)$ is a valid mathematical expression.

Example 1 Find the domain of the function $f(x, y) = \sqrt {1 - x^2 - y^2}$ .
The square root leads to a meaningful expression only if $1 - x^2 - y^2 \geq 0$ . This is equivalent to $x^2 + y^2 \leq 1$ Points in $\R ^2$ satisfying this inequality are either inside or on the unit circle. The domain is thus the closed unit disk.

(Problem 1a) Find the domain of the function $f(x, y) = \frac {x}{\sqrt {x^2 + y^2}}$ .

(Problem 1b) Find the domain of the function $f(x, y) = \frac {1}{y - x}$ .

(Problem 1c) Find the domain of the function $f(x, y) = \ln (xy)$ .

The graph of a function of two variables $z = f(x, y)$ is the set of points in $\R ^3$ of the form $(x, y, f(x,y))$ and the graph is called a surface.

Example 2 Describe the graph of the function $z = \sqrt {1 - x^2 - y^2}$ in $\R ^3$ .
As mentioned in example 1, the domain of $f$ is the set of all points $(x, y)$ in $\R ^2$ such that $x^2 + y^2 \leq 1$ which is the closed unit disc, i.e, the unit circle together with its interior.
To recognize the surface, square both sides of the equation to get $z^2 = 1 - x^2 - y^2$ which is equivalent to $x^2 + y^2 + z^2 = 1$ This is the unit sphere in $\R ^3$ . In the original function, $z\geq 0$ since $z$ is defined using a positive square root. Hence, the graph of $z = f(x, y) = \sqrt {1 - x^2 - y^2}$ is the “northern” hemisphere of the unit sphere (including the equator).

(Problem 2a) Describe the graph of the function $z = x - 2y$ in $\R ^3$ .

(Problem 2b) Describe the graph of the function $z = x^2 + 4y^2$ in $\R ^3$ .

(Problem 2c) Describe the graph of the function $z = \sqrt {x^2 + y^2}$ in $\R ^3$ .

Recall from section 1.8 that the graph of the equation $\frac {x^2}{a^2} + \frac {y^2}{b^2} - \frac {z^2}{c^2} = 0$ is a cone.

1 Level Curves

A level curve for the surface $z = f(x, y)$ is the curve in the $xy$ -plane given by $f(x, y) = k$ for some real number $k$ .
Level curves are the basis for topographical maps like the one below.

In the map above, the contour curves are used to indicate elevation.
When several level curves of $f(x, y)$ are sketched in the same $xy$ -plane, the figure is called a contour map.

Example 3 Describe the level curves of the function $z = f(x, y) = \sqrt {1-x^2 - y^2}$ and sketch a contour map consisting of at least 3 level curves.
The level curves have the form $\sqrt {1 - x^2 - y^2} = k$ for some constant, $k$ . Note that if $k < 0$ then the graph is the empty set.
To recognize the curve in the $xy$ -plane, square both sides and rearrange to get: $x^2 + y^2 = 1- k^2$ If $0 \leq k < 1$ this is a circle in the $xy$ -plane with center at the origin and radius $\sqrt {1-k^2}$ .
If $k = 1$ , this is just the single point $(0, 0)$ and
if $k>1$ then the graph is the empty set.
In the figure below, the chosen values for $k$ are: $\,0, \frac 23, \frac {9}{10}$ and $1$ . The corresponding radii of the circles are: $\,1, \frac {\sqrt {5}}{3}, \frac {\sqrt {19}}{10}$ and $0$ .

(Problem 3a) Describe the level curves of the function $f(x,y) = x - 2y$ and sketch a contour map consisting of at least 3 level curves.

(Problem 3b) Describe the level curves of the function $f(x, y) = x^2 + 4y^2$ and sketch a contour map consisting of at least 3 level curves.

(Problem 3c) Describe the level curves of the function $f(x, y) = \sqrt {x^2 + y^2}$ and sketch a contour map consisting of at least 3 level curves.

(Problem 3d) Describe the level curves of the function $f(x,y) = y - e^x$ and sketch a contour map consisting of at least 3 level curves.

Here is a video solution of problem 3, parts a, b and c:

The graph of a function of one variable, $y= f(x)$ can be thought of as the level curve of the function of two variables given by $g(x, y) = y-f(x)$ corresponding to $k = 0$ .

2 Functions of Three Variables

The volume of a rectangular solid is length $\times$ width $\times$ height, so we can think of the volume as a function of three variables: $Volume = f(length, width, height)$ In general a function of three variables looks like $f(x, y, z)$ . The graph of such a function consists of the set of points in $\R ^4$ of the form $(x, y, z, f(x, y, z))$ . Our humanity limits us to three dimensions or fewer, so we are unable to produce the graph of a function of three variables. However, we can consider their level surfaces.

Example 4 Describe the level surfaces of the function $f(x, y, z) = x^2 + 4y^2 + 9 z^2$ .
The level surfaces are produced by the equation $f(x, y, z) = k$ for constants, $k$ .
In this example, the corresponding equation is $x^2 + 4y^2 + 9z^2 = k$ which is an ellipsoid for $k >0$ , and just the single point $(0, 0, 0)$ for $k = 0$ . For $k<0$ , the level surface is the empty set.

(Problem 4) Describe the level surfaces of the function $f(x, y, z) = x^2 + y^2 - z^2$ .

Try $k = -1, 0, 1$

Refer to the end of section 1.8 for the equations of hyperboloids and cones

The graph of a function of two variables, $z= f(x,y)$ can be thought of as the level surface of the function of three variables given by $g(x, y, z) = z-f(x,y)$ corresponding to $k = 0$ .

2025-03-20 20:05:54