Functions of Several Variables

Multivariate Functions

In one-dimensional calculus, we studied functions that take a single real number as input and produce a single real number as output, i.e., functions of the form:

$f : D \subseteq \mathbb{R} \to \mathbb{R}, \quad x \mapsto f(x)$

where we have a function $f$ that maps elements from a domain $D$ (a subset of the real numbers) to the real numbers. The notation $x \mapsto f(x)$ indicates that the function takes an input $x$ and produces an output $f(x)$ . Here, $x$ is a real number from the domain $D$ , and $f(x)$ is the output of the function for that input.

In multivariate/multi-dimensional calculus, we extend this concept to functions that can take multiple inputs and produce multiple outputs, called multivariate functions or vector-valued functions.

A multivariate function takes a vector of $n$ inputs and produces a vector of $m$ outputs:

$f : D \subseteq \mathbb{R}^n \to \mathbb{R}^m, \quad \mathbf{x} = \begin{pmatrix} x_1 \\ \vdots \\ x_n \end{pmatrix} \mapsto f(\mathbf{x}) = \begin{pmatrix} f_1(x_1, \ldots, x_n) \\ \vdots \\ f_m(x_1, \ldots, x_n) \end{pmatrix}$

where $\mathbf{x}$ is a vector in $\mathbb{R}^n$ (the domain), and $f(\mathbf{x})$ is a vector in $\mathbb{R}^m$ (the codomain — not to be confused with the image, which is the subset of $\mathbb{R}^m$ actually reached by $f$ ). Each component function $f_i$ maps the input vector $\mathbf{x}$ to a single real number, and together they form the output vector in $\mathbb{R}^m$ . The notation $\mathbf{x} \mapsto f(\mathbf{x})$ indicates that the function takes an input vector $\mathbf{x}$ and produces an output vector $f(\mathbf{x})$ .

The one-dimensional case is a special case of this more general framework, where $n = m = 1$ .

To distinguish between scalars and vectors, we will use boldface letters (e.g., $\mathbf{x}$ ) to denote vectors, while regular letters (e.g., $x$ ) will denote scalars. In handwriting, we will denote vectors by underlining the letter (e.g., $\underline{x}$ ) to differentiate them from scalars.

As can be seen above with $\mathbf{x}$ , when we say “vector,” we mean a column vector, which is a common convention in mathematics. A column vector is a matrix with a single column and multiple rows, representing a point in multi-dimensional space.

Common Types of Multivariate Functions

We can define various types of multivariate functions based on the dimensions of their domain and codomain:

Curves

A spaghetti/noodle shape in 2D or 3D space. The input is a single parameter (e.g., time) that traces out a path in space.

$n = 1$ and $m \in \mathbb{N}$

The symbol $\mathbb{N}$ stands for the natural (counting) numbers $\{1, 2, 3, \ldots\}$ . This notation specifies that a curve is traced over a single input dimension ( $n=1$ , representing time), but the resulting path can exist in a 1D, 2D, 3D, or any higher integer-dimensional space because the output dimension $m$ must be a solid counting number.

Plane curves (curves in 2D)

A plane curve takes a single real parameter $t \in D$ and produces a point in 2D space, tracing a path through the plane as $t$ varies:

$\gamma : D \subseteq \mathbb{R} \to \mathbb{R}^2 \quad (n = 1,\ m = 2)$

$\gamma: [0, 2\pi] \to \mathbb{R}^2, \quad \gamma(t) = \begin{pmatrix} 2\cos t \\ \sin t \end{pmatrix}$

In the example above, the function $\gamma$ takes a real number $t$ (which can be thought of as time) and maps it to a point in 2D space. The $x$ -coordinate is given by $2\cos t$ and the $y$ -coordinate by $\sin t$ . As $t$ varies from $0$ to $2\pi$ , the point traces out a curve in the plane, which in this case is an ellipse.

Space curves (curves in 3D)

A space curve takes a single real parameter $t \in D$ and produces a point in 3D space, tracing a path through three-dimensional space as $t$ varies:

$\gamma : D \subseteq \mathbb{R} \to \mathbb{R}^3 \quad (n = 1,\ m = 3)$

$\gamma: [0, 8\pi] \to \mathbb{R}^3, \quad \gamma(t) = \begin{pmatrix} e^{-0.1t}\cos t \\ e^{-0.1t}\sin t \\ t \end{pmatrix}$

In the example above, $\cos t$ and $\sin t$ create circular motion in the $x$ - $y$ plane, while the exponential term $e^{-0.1t}$ causes the radius of that circle to shrink over time — the spiral gets tighter and tighter. The $z$ -coordinate increases linearly with $t$ , so the curve also climbs upward as it spirals.

Surfaces

Surfaces are intrinsically two-dimensional objects that can be embedded in three-dimensional space. For example, the surface of a ball is itself a 2D area embedded in a 3D volume.

A parametric surface is defined by a map $(u, v) \in D \mapsto \mathbb{R}^3$ that takes two free parameters and produces a point in 3D space. As $(u, v)$ varies continuously over the domain $D$ , the output traces out a 2D surface embedded in 3D. Unlike an explicit surface $z = f(x, y)$ — where the surface is directly the graph of a function — a parametric surface uses two auxiliary parameters with no inherent geometric meaning, which makes it possible to describe shapes like spheres or tori that cannot be expressed as a single function of $x$ and $y$ .

$\phi : D \subseteq \mathbb{R}^2 \to \mathbb{R}^3 \quad (n = 2,\ m = 3)$

The two input parameters act like coordinates on the surface itself — by varying them over the domain $D$ , we trace out every point on the surface.

$\phi: [0, 2\pi] \times [0, 2] \to \mathbb{R}^3, \quad \phi(u,v) = \begin{pmatrix} \cos u \\ \sin u \\ v \end{pmatrix}$

In the example above, the function $\phi$ takes two parameters $u$ and $v$ and maps them to a point in 3D space. The domain is given by the Cartesian product of the intervals $[0, 2\pi]$ and $[0, 2]$ . This notation indicates a combination of all possible inputs: $u$ can be any real number from $0$ to $2\pi$ and $v$ can independently be any real number from $0$ to $2$ . Rather than representing just four corner points, this domain forms a solid, filled-in 2D rectangular “piece of paper.” The function then rolls this rectangle into a 3D cylinder. The circular cross-section is mapped out by $\cos u$ and $\sin u$ , creating a default unit radius of 1, making $u$ the angle parameter while the height of the cylinder is directly mapped from $v$ .

The radius of the cylinder is constant (equal to 1) in this example because $\cos u$ and $\sin u$ are bounded by 1 and not multiplied by any expanding factor. However, we can create more complex surfaces by allowing the radius to vary with $u$ and $v$ . For instance, in the next example, we have a surface where the radius changes based on the cosine of $u$ , creating a wavy pattern around the cylinder.

$\phi: [0, 2\pi] \times [0, 2\pi] \to \mathbb{R}^3, \quad \phi(u,v) = \begin{pmatrix} (2+\cos u)\cos v \\ (2+\cos u)\sin v \\ \sin u \end{pmatrix}$

Scalar Fields

Assigns a single real number to each point in space. Intuitively, imagine walking around a room with a thermometer or a light meter: at every specific coordinate $(x, y, z)$ in the room, there is exactly one temperature or brightness value. Every point is mapped to a single scalar.

A scalar field assigns a single real number to each point in an $n$ -dimensional space:

$f : D \subseteq \mathbb{R}^n \to \mathbb{R} \quad (n \in \mathbb{N},\ m = 1)$

$f: [-4,4] \times [-5,5] \to \mathbb{R}, \quad f(x,y) = 2(x^2+y^2)$

$f: \mathbb{R}^2 \to \mathbb{R}, \quad f(x,y) = xy\exp^{-(x^2+y^2)}$

Vector Fields

Assigns a vector to each point in space. Intuitively, imagine holding a small wind-vane at every coordinate in a room to measure airflow. A single number isn’t enough; you need multiple numbers to describe both the speed and the direction of the wind. The function effectively attaches a directional arrow to every single point in that space.

A vector field assigns a vector to each point in space:

$\mathbf{v} : D \subseteq \mathbb{R}^n \to \mathbb{R}^n \quad (n = m,\ n \in \mathbb{N})$

The strict requirement for vector fields is that the output must live in the exact same dimensionality space as the input ( $n = m$ ), meaning every point in the domain gets an arrow pointing somewhere within that same geometric space.

$\mathbf{v}: \mathbb{R}^2 \to \mathbb{R}^2, \quad \mathbf{v}(x,y) = \begin{pmatrix} -y \\ x \end{pmatrix}$

$\mathbf{v}: \mathbb{R}^3 \to \mathbb{R}^3, \quad \mathbf{v}(x,y,z) = \begin{pmatrix} z \\ y \\ -x \end{pmatrix}$

Topology in Higher Dimensions

The familiar notions of open, closed, and half-open intervals from 1D calculus — $(a,b)$ , $(a,b]$ , $[a,b)$ , $[a,b]$ — generalize naturally to higher-dimensional spaces.

Complement of a Set

Let $D \subseteq \mathbb{R}^n$ be some domain. Its complement in $\mathbb{R}^n$ is

$D^c = \mathbb{R}^n \setminus D.$

Euclidean Norm

To state the remaining definitions precisely, we need two building blocks: the Euclidean norm and the notion of an $\varepsilon$ -ball.

The Euclidean norm $\|\cdot\|$ is the standard “ruler distance” — it measures the straight-line length (magnitude) from the origin to a coordinate point $\mathbf{x} = (x_1, \ldots, x_n)^\top \in \mathbb{R}^n$ :

$\|(x_1, \ldots, x_n)^\top\| = \sqrt{x_1^2 + \cdots + x_n^2}$

In $\mathbb{R}^2$ this is just the Pythagorean theorem: the distance from the origin to the point $(x_1, x_2)$ is $\sqrt{x_1^2 + x_2^2}$ .

The ε-Ball

The $\varepsilon$ -ball of radius $\varepsilon > 0$ centered at $\mathbf{x}_0$ is the set of all points within distance $\varepsilon$ of $\mathbf{x}_0$ :

$B_\varepsilon(\mathbf{x}_0) = \{\, \mathbf{x} \in \mathbb{R}^n \mid \|\mathbf{x} - \mathbf{x}_0\| < \varepsilon \,\}$

Think of an $\varepsilon$ -ball as a “bubble” or a “safety zone” drawn around a specific center point. It is the higher-dimensional analogue of an open interval: in $\mathbb{R}^1$ it reduces to the line segment $(x_0 - \varepsilon,\, x_0 + \varepsilon)$ , in $\mathbb{R}^2$ it is a flat, filled-in circular disk, and in $\mathbb{R}^3$ it is a solid 3D sphere. Crucially, because the formula uses a strict inequality ( $< \varepsilon$ , not $\leq \varepsilon$ ), the boundary or the “peel” of the ball (enclosing surface) is explicitly excluded. This missing peel is exactly what makes it an open ball.

Inner Points

A point $\mathbf{x}_0 \in D$ is called an inner point of $D$ if there exists some $\varepsilon > 0$ such that $B_\varepsilon(\mathbf{x}_0) \subseteq D$ — that is, we can place an open ball around $\mathbf{x}_0$ that fits entirely within $D$ , without touching or crossing its boundary.

Because real number spaces are perfectly continuous and not constructed from fixed pixels, there is no such thing as a “closest point” to a boundary. If you pick a point near an excluded boundary, there is always a microscopic gap between your point and the edge. Therefore, you can always choose an $\varepsilon$ small enough so that your $\varepsilon$ -ball squeezes perfectly into that gap without spilling outside the domain.

Interior

The set of all inner points of $D$ is called the interior of $D$ , denoted $\mathring{D}$ .

Open Sets

A set $D \subseteq \mathbb{R}^n$ is called open if it equals its own interior: $D = \mathring{D}$

In plain terms: every point in $D$ is an inner point. There are no “edge” points that belong to $D$ — every point inside has some breathing room, meaning you can move a little in any direction and still stay inside $D$ .

Boundary Points

A point $\mathbf{x}_0 \in \mathbb{R}^n$ is called a boundary point of $D$ if every open ball around it, no matter how small, contains points both inside and outside $D$ :

$B_\varepsilon(\mathbf{x}_0) \cap D \neq \emptyset \quad \text{and} \quad B_\varepsilon(\mathbf{x}_0) \cap D^c \neq \emptyset \qquad \text{for all } \varepsilon > 0.$

In other words, no matter how small we make $\varepsilon$ , the ball $B_\varepsilon(\mathbf{x}_0)$ will always be partially inside and partially outside $D$ — we can shrink it as much as we want, and it will still cross the edge. This is possible because we are working in $\mathbb{R}^n$ : we can always zoom in further, there is no limit to how small a ball we can draw. No matter how close we look, $\mathbf{x}_0$ is always right on the edge, and therefore can never be an inner point.

Note that a boundary point need not belong to $D$ itself — it is simply a point that straddles (has one foot inside and one foot outside) the edge of $D$ , and may lie either inside or outside it.

In $\mathbb{R}$ , the open interval $D = (0, 1)$ is an open set. Take any point $x \in (0, 1)$ — it sits somewhere strictly between $0$ and $1$ , so it has some breathing room on both sides. Setting

$\varepsilon = \min(x,\ 1 - x)$

picks the smaller of the two distances to the endpoints, guaranteeing that the ball $B_\varepsilon(x) = (x - \varepsilon,\ x + \varepsilon)$ stays fully inside $(0, 1)$ without poking out either side. Since we can do this for every $x \in D$ , every point is an inner point, confirming that $D$ is open.

The endpoints $0$ and $1$ are boundary points: any ball around them, no matter how small, reaches into $(0, 1)$ on one side and outside it on the other — they sit right on the edge. Note that neither $0$ nor $1$ belongs to $D$ , which is consistent with $D$ being open.

Boundary

The set of all boundary points of $D$ is called the boundary of $D$ , denoted $\partial D$ .

The boundary represents the outer geometric shell of a shape. It is often intuitively referred to as the “peel” or the “hull” (frequently used by German mathematicians translating the word Hülle).

Closure

The closure of $D$ is the set $\bar{D} = D \cup \partial D$ — that is, $D$ together with all of its boundary points.

Conceptually, taking the closure is the mathematical command to take a domain $D$ and forcefully glue its entire peel back onto it. It “seals off” $D$ by including every point that sits on its edge, regardless of whether those points were originally in $D$ or not. If $D$ is already a closed set, its peel is already there, meaning the closure operation does absolutely nothing to it. If $D$ is open, the closure plugs the holes at the boundary.

Closed Sets

A set $D \subseteq \mathbb{R}^n$ is called closed if it contains all of its boundary points: $\partial D \subseteq D$ , or equivalently, $\bar{D} = D$ .

A closed set is the complement of an open set — and conversely, an open set is the complement of a closed set. Note that “open” and “closed” are not opposites — they are independent properties a set may or may not have. A set can be:

both open and closed: $\mathbb{R}^n$ itself (no boundary at all) and $\emptyset$ (vacuously satisfies both definitions),
neither open nor closed: the half-open interval $[0, 1)$ in $\mathbb{R}$ — it contains one boundary point ( $0$ ) but not the other ( $1$ ), so it fails both definitions.

The closed interval $D = [0, 1]$ in $\mathbb{R}$ is a closed set. Its boundary points are $0$ and $1$ , and both belong to $D$ . Compare this with the open interval $(0, 1)$ : same boundary $\{0, 1\}$ , but neither endpoint belongs to the set — so $(0, 1)$ is open, not closed.

Bounded Sets

A set $D \subseteq \mathbb{R}^n$ is called bounded if there exists some $K \in \mathbb{R}$ such that $\|\mathbf{x}\| < K$ for all $\mathbf{x} \in D$ .

Recall that $\|\mathbf{x}\|$ is the Euclidean distance from the origin to the point $\mathbf{x}$ . So boundedness simply says: every point in $D$ lies within distance $K$ of the origin — equivalently, $D$ fits entirely inside some $\varepsilon$ -ball $B_K(\mathbf{0})$ of finite radius.

Intuitively, a shape is bounded if you can construct a giant, finite box or cage that completely encloses it. No vector in $D$ can be “infinitely long” or shoot off to infinity in any direction. A set is unbounded if it escapes any cage you try to build around it, stretching off infinitely like a laser beam, a standard parabola, or the entire real line $\mathbb{R}$ .

Compact Sets

A set $D \subseteq \mathbb{R}^n$ is called compact if it is both closed and bounded.

Convex Sets

A set $D \subseteq \mathbb{R}^n$ is called convex if for any two points $\mathbf{x}, \mathbf{y} \in D$ , the entire line segment connecting them also lies in $D$ :

$\forall\, \mathbf{x}, \mathbf{y} \in D :\ \alpha\mathbf{x} + (1 - \alpha)\mathbf{y} \in D \qquad \text{for all } 0 \leq \alpha \leq 1.$

The expression $\alpha\mathbf{x} + (1 - \alpha)\mathbf{y}$ is called a convex combination of $\mathbf{x}$ and $\mathbf{y}$ . As the scalar $\alpha$ varies continuously from $1$ to $0$ , this mathematical formula traces the straight line segment connecting the two points: at $\alpha = 1$ you are exactly at $\mathbf{x}$ , at $\alpha = 0$ you are at $\mathbf{y}$ , and at $\alpha = \tfrac{1}{2}$ you are precisely at the midpoint.

Convexity mathematically dictates that for any two points chosen inside $D$ , this entire connecting line segment must remain completely within $D$ . Intuitively, this can be visualized as a “line-of-sight” test: if a domain is convex, you can stand at any coordinate inside it, and a friend can stand at any other coordinate, and you will always have a direct, unobstructed line of sight to each other that never crosses outside the boundary. Consequently, a convex shape has no “dents”, “holes”, or “concave bays.”

A filled disk $\{\mathbf{x} \in \mathbb{R}^2 \mid \|\mathbf{x}\| \leq r\}$ is convex: any straight line between two points inside the disk stays inside the disk. A crescent moon, a donut shape, or an “L-shaped” room, on the other hand, is not convex — one can easily find two points within the shape whose connecting line segment passes through the empty space outside of it.

Continuity

Continuity on multi-dimensional domains is defined via the convergence of vector sequences.

Convergence of Vector Sequences

In $\mathbb{R}^n$ , we work with vector sequences of the form $\left(\mathbf{x}^{(k)}\right)_{k \in \mathbb{N}_0}$ , where each element is a vector:

$\mathbf{x}^{(k)} = (x_1^{(k)}, \ldots, x_n^{(k)})^\top \in \mathbb{R}^n \quad \text{for all } k \in \mathbb{N}_0$

Convergence of a Vector Sequence

A sequence of vectors $\left(\mathbf{x}^{(k)}\right)_{k \in \mathbb{N}_0}$ converges to the limit vector $\mathbf{x}$ if the distance between $\mathbf{x}^{(k)}$ and $\mathbf{x}$ vanishes as $k \to \infty$ :

$\lim_{k \to \infty} \| \mathbf{x}^{(k)} - \mathbf{x} \| = 0$

In other words, a vector sequence $\left(\mathbf{x}^{(k)}\right)$ converges to $\mathbf{x}$ if and only if the real-valued sequence of distances $\| \mathbf{x}^{(k)} - \mathbf{x} \|$ converges to zero — reducing vector convergence to the familiar notion of scalar convergence.

We can express this convergence in several ways depending on how formal the context requires. The following are all equivalent ways to write the exact same limit behavior:

$\mathbf{x}^{(k)} \xrightarrow{k \to \infty} \mathbf{x} \qquad \mathbf{x}^{(k)} \to \mathbf{x} \qquad \lim_{k \to \infty} \mathbf{x}^{(k)} = \mathbf{x}$

The first notation is explicit, detailing precisely what variable is moving to infinity. The second is a common shorthand adopted when $k \to \infty$ is mathematically obvious from the context. The third is the classic, highly formal algebraic limit notation.

Convergence of a vector sequence is equivalent to convergence of all its components simultaneously — none of them can diverge. Formally:

$\mathbf{x}^{(k)} = \begin{pmatrix} x_1^{(k)} \\ \vdots \\ x_n^{(k)} \end{pmatrix} \to \mathbf{x} = \begin{pmatrix} x_1 \\ \vdots \\ x_n \end{pmatrix} \iff x_1^{(k)} \to x_1, \ldots, x_n^{(k)} \to x_n$

Continuity of Vector Functions

Given a vector function $f : D \subseteq \mathbb{R}^n \to \mathbb{R}^m$ , we say that $f$ is:

continuous at $\mathbf{a} \in D$ if, for every sequence $\left(\mathbf{x}^{(k)}\right)_{k \in \mathbb{N}_0}$ in $D$ with $\mathbf{x}^{(k)} \to \mathbf{a}$ , the corresponding output sequence $\left(f\left(\mathbf{x}^{(k)}\right)\right)_{k \in \mathbb{N}_0}$ in $\mathbb{R}^m$ converges to $f(\mathbf{a})$ . In short, there is no “teleporting” allowed. As the input smoothly approaches a specific destination $\mathbf{a}$ , the output must smoothly approach the actual output at that destination $f(\mathbf{a})$ without sudden jumps or glitches.
continuous on $D$ if it is continuous at every point $\mathbf{a} \in D$ .

Continuity in multi-dimensional spaces is equivalent to componentwise continuity. A vector function

$f : D \subseteq \mathbb{R}^n \to \mathbb{R}^m, \quad \begin{pmatrix} x_1 \\ \vdots \\ x_n \end{pmatrix} \mapsto \begin{pmatrix} f_1(x_1, \ldots, x_n) \\ \vdots \\ f_m(x_1, \ldots, x_n) \end{pmatrix}$

is effectively a stack of scalar fields. $f$ is continuous at $\mathbf{a}$ or on $D$ if and only if each individual component scalar field

$f_i : D \to \mathbb{R}, \quad (x_1, \ldots, x_n)^\top \mapsto f_i(x_1, \ldots, x_n), \quad i \in \{1, \ldots, m\}$

is properly continuous at $\mathbf{a}$ or on $D$ . Think of a drone flying through the air: for the drone’s overall 3D trajectory to be continuous, its left/right movement, its forward/backward movement, and its up/down movement must all be perfectly continuous. If even a single component teleports, the whole trajectory breaks.

The standard continuity-preservation rules from 1D carry over directly to multi-dimensional functions. If $f(\mathbf{x})$ and $g(\mathbf{x})$ are continuous on $D$ , so are all functions built from them via the usual operations.

Suppose $f(\mathbf{x})$ and $g(\mathbf{x})$ are continuous on $D$ . Then the following composite operations produce new functions that are also continuous on $D$ :

Linear combination: $5f(\mathbf{x}) - 2g(\mathbf{x})$
Product: $f(\mathbf{x})\, g(\mathbf{x})$
Composition: $f(g(\mathbf{x}))$
Division: $\dfrac{f(\mathbf{x})}{g(\mathbf{x})}$

For division, there is one strict condition: the divisor function on the bottom must be non-zero ( $g(\mathbf{x}) \neq 0$ ) for all $\mathbf{x} \in D$ . If the denominator hits exactly zero at any point, the fraction explodes to infinity, generating a mathematical hole or vertical jump that immediately destroys the function’s continuity.