This is a beta course, so its structure, chapters, and examples may continue to change.

Basic Concepts of Multivariable Functions

Definition of Multivariable Functions

Functions of Two Variables

Function of Two Variables

Let $D$ be a set of points in the plane. If for every point $(x, y)$ in $D$ , a definite rule $f$ assigns a unique real number $z$ , then $z$ is called a function of two variables:

$z = f(x, y), \quad (x, y) \in D$

where $D$ is the domain, $z$ the dependent variable, and $x, y$ the independent variables. The set of all possible values of $z$ is the range.

Functions of Three or More Variables

Similarly, a function of three variables is written

$w = f(x, y, z), \quad (x, y, z) \in D$

where $D$ is a set of points in 3-space. In general, an n-variable function is

$u = f(x_1, x_2, \ldots, x_n), \quad (x_1, x_2, \ldots, x_n) \in D$

with $D$ a set of points in $\mathbb{R}^n$ .

Moving from one variable to several is not just adding more inputs—the domain changes from an interval to a multi-dimensional region, which fundamentally changes limit, continuity, and differentiability. For example, a single-variable limit only considers two directions (left and right), while a two-variable limit must consider infinitely many approaching paths.

Geometric Meaning

Geometric interpretation

A two-variable function $z = f(x, y)$ : represents a surface in 3-space; the domain $D$ is the projection of the surface onto the $xOy$ -plane.
A three-variable function $w = f(x, y, z)$ : cannot be drawn directly in 3-space; it is a hypersurface in 4-space, often visualized through level surfaces $f(x,y,z) = c$ .
An n-variable function: a hypersurface in $n$ -space.

Level Curves and Level Surfaces

For $z = f(x,y)$ , a level curve (contour line) is defined by

$f(x, y) = c$

where $c$ is a constant. Points on the same level curve share the same function value; topographic contour maps are the classic example.

For $w = f(x,y,z)$ , a level surface is defined by

$f(x, y, z) = c$

Isobaric and isothermal surfaces in meteorology are level surfaces.

Finding the Domain

Like the one-variable case, the domain consists of all inputs for which the expression is defined. Common restrictions:

Function type	Restriction
Fraction $\frac{1}{f}$	$f \neq 0$
Even root $\sqrt[2n]{f}$	$f \geq 0$
Logarithm $\log_a f$	$f > 0$
Arcsine/arccosine $\arcsin f / \arccos f$	$
Tangent $\tan f$	$f \neq k\pi + \frac{\pi}{2}$

For functions arising from real problems, the domain is also restricted by practical meaning.

Example 1: Domain of a Two-Variable Function

Find the domain of $f(x, y) = \sqrt{1 - x^2 - y^2}$ .

Answer and Explanation(4 个标签)

domaintwo variablesrootclosed disk

Approach: an even root requires the radicand to be nonnegative.

Detailed steps:

Require $1 - x^2 - y^2 \geq 0$
That is $x^2 + y^2 \leq 1$
The domain is $D = \{(x, y) \mid x^2 + y^2 \leq 1\}$

Answer: the closed disk centered at the origin with radius 1 (including the boundary).

Example 2: Domain of a Compound Function

Find the domain of $f(x, y) = \ln(x^2 + y^2 - 1) + \frac{1}{\sqrt{4 - x^2 - y^2}}$ .

Answer and Explanation(5 个标签)

domaincompound functionlogarithmrootannulus

Approach: consider the restrictions from the logarithm and the root separately, then intersect.

Detailed steps:

Logarithm: $x^2 + y^2 - 1 > 0$ , i.e. $x^2 + y^2 > 1$
Root: $4 - x^2 - y^2 > 0$ , i.e. $x^2 + y^2 < 4$
Together: $1 < x^2 + y^2 < 4$
The domain is $D = \{(x, y) \mid 1 < x^2 + y^2 < 4\}$

Answer: the open annulus with inner radius 1 and outer radius 2 (boundaries excluded).

Limits of Multivariable Functions

Definition of the Double Limit

Double Limit

Let $f(x, y)$ be defined on a deleted neighborhood of $P_0(x_0, y_0)$ . If for every $\varepsilon > 0$ there exists $\delta > 0$ such that whenever $0 < \sqrt{(x - x_0)^2 + (y - y_0)^2} < \delta$ , we have $|f(x, y) - A| < \varepsilon$ , then $A$ is called the double limit of $f$ as $(x, y) \to (x_0, y_0)$ :

$\lim_{(x, y) \to (x_0, y_0)} f(x, y) = A$

Key difference: a one-variable limit has only two directions, but a two-variable limit requires that $P$ approaches $P_0$ in every way possible and $f$ still approaches the same constant $A$ . This is why multivariable limits are fundamentally harder.

Properties of Limits

定理1

Uniqueness: if the limit exists, it is unique.
Local boundedness: if the limit exists, the function is bounded in some deleted neighborhood.
Arithmetic: the limit of a sum/difference/product/quotient equals the corresponding combination of limits (provided the denominator limit is nonzero).
Squeeze theorem: if $g \leq f \leq h$ and $\lim g = \lim h = A$ , then $\lim f = A$ .
Composition: if $\lim f = A$ and $\lim_{u\to A} g(u) = B$ , then $\lim g(f) = B$ .

Iterated Limits

Iterated Limit

Fix one variable while the other approaches, then let the fixed one approach—the result is an iterated limit:

$\lim_{x \to x_0} \lim_{y \to y_0} f(x, y), \quad \lim_{y \to y_0} \lim_{x \to x_0} f(x, y)$

Relationship between double and iterated limits:

If the double limit exists, the iterated limits need not exist; but if both iterated limits exist, they must be equal and equal to the double limit.
Both iterated limits existing and being equal does not guarantee the double limit exists.
If the two iterated limits exist but differ, the double limit definitely does not exist.
Iterated limits cannot be exchanged at will.

Proving a Limit Does Not Exist

To show a limit fails to exist:

Path method: find two different paths with different limits (commonly $y=kx$ , $y=kx^2$ ).
Iterated limit method: the two iterated limits exist but differ.
Polar method: set $x=\rho\cos\theta, y=\rho\sin\theta$ ; if the limit depends on $\theta$ , it does not exist.

To show a limit exists: squeeze theorem, polar transformation independent of $\theta$ , equivalent infinitesimals, or substitution.

Example 3: A Nonexistent Limit (Straight Paths)

Show that $\lim_{(x, y) \to (0, 0)} \frac{xy}{x^2 + y^2}$ does not exist.

Answer and Explanation(3 个标签)

nonexistent limitpath methodclassic counterexample

Detailed steps:

Along the $x$ -axis ( $y=0$ ): $\lim_{x\to0} \frac{0}{x^2} = 0$
Along $y=x$ : $\lim_{x\to0} \frac{x^2}{2x^2} = \frac{1}{2}$
The two paths give different limits, so the limit does not exist.

Answer: the limit does not exist.

Example 4: A Nonexistent Limit (Parabolic Path)

Show that $\lim_{(x, y) \to (0, 0)} \frac{x^2 y}{x^4 + y^2}$ does not exist.

Answer and Explanation(3 个标签)

nonexistent limitparabolic pathhigher-order infinitesimal

Detailed steps:

Along the $x$ -axis: the limit is $0$
Along $y=x^2$ : $\lim_{x\to0} \frac{x^4}{x^4+x^4} = \frac{1}{2}$
The limits differ, so the limit does not exist.

Even if the limit along every straight line $y=kx$ were $0$ , the double limit could still fail to exist. Every path must be considered.

Answer: the limit does not exist.

Example 5: Computing a Limit (Squeeze / Polar)

Find $\lim_{(x, y) \to (0, 0)} \frac{x^2 y}{x^2 + y^2}$ .

Answer and Explanation(3 个标签)

existing limitsqueeze theorempolar coordinates

Detailed steps:

Method 1 — squeeze theorem: $0 \leq \left|\frac{x^2 y}{x^2+y^2}\right| \leq \frac{(x^2+y^2)\sqrt{x^2+y^2}}{x^2+y^2} = \sqrt{x^2+y^2} \to 0$ so the limit is $0$ .

Method 2 — polar coordinates: put $x=\rho\cos\theta,\ y=\rho\sin\theta$ . Then $\frac{x^2 y}{x^2+y^2}=\rho\cos^2\theta\sin\theta\to0$ as $\rho\to0$ , independent of $\theta$ .

Answer: the limit is $0$ .

Continuity of Multivariable Functions

Definition

Continuity of a Two-Variable Function

Let $f(x, y)$ be defined on a neighborhood of $P_0(x_0, y_0)$ . If

$\lim_{(x, y) \to (x_0, y_0)} f(x, y) = f(x_0, y_0)$

then $f$ is continuous at $P_0$ . If $f$ is continuous at every point of a region $D$ , it is continuous on $D$ .

Equivalently, $f$ is continuous at $P_0$ exactly when:

$f(P_0)$ is defined
$\lim_{P\to P_0} f(P)$ exists
the limit equals the function value

Discontinuities

Point of Discontinuity

If $f(x, y)$ is not continuous at $P_0$ , then $P_0$ is called a point of discontinuity. There are three situations:

$f(P_0)$ is undefined
$\lim_{P\to P_0} f(P)$ does not exist
the limit exists but differs from $f(P_0)$

Unlike the one-variable case, discontinuities of a two-variable function can be isolated points or entire curves. For example, $f(x,y) = \frac{1}{x^2+y^2-1}$ is discontinuous at every point of the unit circle $x^2+y^2=1$ .

Properties of Continuous Functions

定理2

Arithmetic: sums, differences, products, and quotients (with nonzero denominator) of continuous functions are continuous.
Composition: a composition of continuous functions is continuous.
Elementary functions: multivariable elementary functions are continuous on their domains.
Extreme value theorem: a continuous function on a closed bounded region attains a maximum and a minimum.
Intermediate value theorem: on a closed bounded region, a continuous function attains every value between its minimum $m$ and maximum $M$ .
Boundedness theorem: a continuous function on a closed bounded region is bounded.

Uniform Continuity

If for every $\varepsilon > 0$ there exists $\delta > 0$ such that for any two points $P_1, P_2$ in $D$ with $|P_1P_2| < \delta$ , we have $|f(P_1) - f(P_2)| < \varepsilon$ , then $f$ is uniformly continuous on $D$ .

定理3

Cantor’s theorem: a function continuous on a closed bounded region is uniformly continuous there.

Continuity is a local concept (the $\delta$ may depend on the point); uniform continuity is a global concept (one $\delta$ works for the whole region). Uniform continuity implies continuity, but not conversely—for example $f(x,y)=\frac{1}{x}$ is continuous but not uniformly continuous on $(0,1)\times\mathbb{R}$ .

Example 6: Continuity at the Origin

Determine the continuity of $f(x, y) = \begin{cases} \frac{xy}{x^2 + y^2}, & (x,y) \neq (0,0) \\ 0, & (x,y) = (0,0) \end{cases}$ at $(0,0)$ .

Answer and Explanation(3 个标签)

continuitydiscontinuitynonexistent limit

Detailed steps:

From the earlier limit example, $\lim_{(x,y)\to(0,0)} \frac{xy}{x^2+y^2}$ does not exist
Yet $f(0,0) = 0$
The limit does not exist, so the function is discontinuous at $(0,0)$ .

Answer: discontinuous at $(0,0)$ .

Example 7: Extending a Function to Be Continuous

Let $f(x, y) = \frac{x^2 y}{x^2 + y^2}$ for $(x,y)\neq(0,0)$ . Can we define $f(0,0)$ so that $f$ becomes continuous at the origin?

Answer and Explanation(3 个标签)

continuityextensionremovable discontinuity

Detailed steps:

From Example 5, $\lim_{(x,y)\to(0,0)} \frac{x^2 y}{x^2+y^2} = 0$
Define $f(0,0) = 0$
Then the limit equals the function value, so $f$ is continuous at the origin.

Answer: yes—set $f(0,0) = 0$ ; the origin is a removable discontinuity.

Summary

Symbols Used in This Article

Symbol	Type	Reading/Explanation	Meaning in This Article
$D$	point set / region	domain	The domain of the function
$f(x,y)$	two-variable function	function of two variables	The expression of a two-variable function
$z$	dependent variable	dependent variable	The output of a two-variable function
$\mathbb{R}^n$	n-space	n-dimensional space	The space containing the domain
$c$	constant	constant	The value defining a level curve/surface
$\varepsilon, \delta$	Greek letters	epsilon, delta	Used in the limit/continuity definitions
$\rho, \theta$	Greek letters	rho, theta	Polar coordinates

English–Chinese Glossary

English term	Chinese term	Phonetic	Explanation
multivariable function	多元函数	/ˌmʌltiˈvɛriəbəl ˈfʌŋkʃən/	A function of several variables
function of two variables	二元函数	/ˈfʌŋkʃən əv tuː ˈvɛriəbəlz/	A function with two inputs
function of n variables	n元函数	/ˈfʌŋkʃən əv ɛn ˈvɛriəbəlz/	A function with $n$ inputs
domain	定义域	/doʊˈmeɪn/	The set of allowed inputs
range	值域	/reɪndʒ/	The set of output values
level curve	等值线	/ˈlɛvəl kɜːrv/	A curve of constant function value
level surface	等值面	/ˈlɛvəl ˈsɜːrfɪs/	A surface of constant function value
double limit	二重极限	/ˈdʌbəl ˈlɪmɪt/	The limit of a two-variable function
iterated limit	累次极限	/ˈɪtəreɪtɪd ˈlɪmɪt/	A limit taken one variable at a time
continuity	连续性	/ˌkɒntɪˈnjuːəti/	The function value equals its limit
uniform continuity	一致连续	/ˈjuːnɪfɔːm ˌkɒntɪˈnjuːəti/	A global continuity condition
hypersurface	超曲面	/ˈhaɪpərˌsɜːrfɪs/	A surface in higher dimensions

Partial Derivatives and Total Differentials

Learn the definition and computation of partial derivatives, higher-order partials, and the total differential, including the relation between differentiability, partial derivatives, and continuity.