Multivariable Functions — Mathematics Knowledge System

Overview

A multivariable function takes more than one input variable and produces an output — most commonly f(x,y) or f(x,y,z). Instead of a curve, the graph of a two-variable function z=f(x,y) is a surface in 3D space. Multivariable functions are how calculus models anything depending on multiple independent quantities at once: temperature depending on position, profit depending on price and quantity, or a neural network's loss depending on thousands of weights.

Intuition

A single-variable function is a machine with one dial and one readout. A multivariable function is a machine with several dials (inputs) and one readout (output) — turning any dial can change the output, and the interesting behavior is how the output responds to combinations of dial settings, not just one at a time. Visually, f(x,y) traces out a landscape: hills, valleys, and saddle passes, where 'height' at each (x,y) location is the function's output.

Formal Definition

Definition

A real-valued function of n variables maps a point in ℝⁿ to a real number:

f: D \subseteq \mathbb{R}^n \to \mathbb{R}, \qquad (x_1, x_2, \ldots, x_n) \mapsto f(x_1,\ldots,x_n)

D is the domain, a subset of n-dimensional space

\lim_{(x,y)\to(a,b)} f(x,y) = L \iff \forall \varepsilon>0\ \exists \delta>0: 0<\sqrt{(x-a)^2+(y-b)^2}<\delta \implies |f(x,y)-L|<\varepsilon

The limit must agree along EVERY path of approach to (a,b), not just along axes

Multivariable limit

Notation

Notation	Meaning
$f(x,y)$	A function of two real variables
$f(x,y)=k$	The set of points where f equals a constant k — a contour line
$D \subseteq \mathbb{R}^n$	The subset of n-space on which f is defined

Properties

Path-dependence of limits

\text{If } f(x,y) \to L_1 \text{ along one path and } L_2\neq L_1 \text{ along another, the limit does not exist}

Condition: Unlike single-variable limits, there are infinitely many directions of approach to check

Continuity

f \text{ is continuous at } (a,b) \iff \lim_{(x,y)\to(a,b)} f(x,y) = f(a,b)

Level curves / contour plots

f(x,y) = k \text{ for various constants } k \text{ describe the surface's 'topographic map'}

Applications

Temperature T(x,y,z), pressure, and electric potential fields are all multivariable functions of spatial position.

3D Visualization

Worked Examples

The expression under the square root must be non-negative.
$9 - x^2 - y^2 \geq 0 \iff x^2+y^2 \leq 9$

Answer: The closed disk of radius 3 centered at the origin: {(x,y) : x²+y² ≤ 9}

Practice Problems

Difficulty 4/10

Describe the level curves of f(x,y) = x² + y².

Difficulty 5/10

Find the domain of f(x,y) = ln(y − x²).

Difficulty 5/10

A terrain elevation is modelled by z = f(x, y). What do the LEVEL CURVES f(x,y) = c represent, and where have you seen them?

Common Mistakes

Common Mistake

Checking a multivariable limit only along the x-axis and y-axis and concluding it exists.

Agreement along the two coordinate axes is not sufficient — the limit must agree along EVERY possible path of approach, including diagonal lines (y=mx) and curves (y=x²). Checking just two paths can miss a path where the limit disagrees.

Common Mistake

Confusing a level curve f(x,y)=k with the graph of f.

The graph of f is a surface in 3D (z=f(x,y)); a level curve is a 2D curve in the xy-plane showing where the surface has constant height k — like a single contour line on a topographic map, not the whole map.

Quiz

A function z = f(x, y) of two variables can be visualized as:

a curve in the planea surface in 3-D (a height above each point of the xy-plane)a single numbera vector field

The level curves f(x,y) = c of an elevation function are exactly:

gradient vectorscontour lines on a topographic maptangent planespartial derivatives

Summary

A multivariable function f(x₁,...,xₙ) maps a point in n-space to a single real output.
The graph of f(x,y) is a surface in 3D; level curves f(x,y)=k are 2D contour slices.
Multivariable limits must agree along every path of approach, not just along coordinate axes — this is much stricter than single-variable limits.
Domains are regions in ℝⁿ, often defined by inequalities (e.g. under a square root or logarithm).
Foundational for optimization, physics fields, economics, and machine learning loss functions of many variables.

References

WebsiteWikipedia — Function of several real variables