Multivariable Functions

Core Definition

A function of two variables $f: D \subset \mathbb{R}^2 \to \mathbb{R}$ is a mathematical rule that assigns to each ordered pair of real numbers $(x, y)$ in a set $D$ (the domain) exactly one real number, denoted by $z = f(x, y)$ .

More generally, a real-valued function of $n$ variables $f: D \subset \mathbb{R}^n \to \mathbb{R}$ maps an ordered $n$ -tuple of real numbers $(x_1, x_2, \dots, x_n)$ to a single real scalar output $w = f(x_1, x_2, \dots, x_n)$ .

Concept Explanation

From Single-Variable to Multivariable Inputs

In single-variable calculus, functions $y = f(x)$ map a 1D input along the $x$ -axis to a 1D output along the $y$ -axis, generating a 2D planar curve. Real physical and engineering systems, however, rarely depend on a single parameter. Temperature in a room depends on position $(x, y, z)$ ; pressure in a reaction vessel depends on temperature $T$ and volume $V$ ; aerodynamic lift depends on airspeed $v$ , air density $\rho$ , wing area $A$ , and angle of attack $\alpha$ .

Single-Variable:     x in R  ------[ f ]------>  y in R         (Graph: 2D Curve)
Two-Variable:   (x, y) in R^2 -----[ f ]------>  z in R         (Graph: 3D Surface)
Three-Variable: (x,y,z) in R^3 ----[ f ]------>  w in R         (Graph: 4D Hypersurface)

Geometric Graph of a Function of Two Variables

The graph of a function of two variables $f(x, y)$ is the set of all points $(x, y, z) \in \mathbb{R}^3$ such that $z = f(x, y)$ and $(x, y) \in D$ . Geometrically, this graph forms a two-dimensional surface hovering over the domain region $D$ in the $xy$ -plane.

                    z ^
                      |          * (x, y, f(x,y))  <-- Point on 3D Surface
                      |         /|
                      |        / |
                      |       /  | z = f(x, y)
                      |      /   |
                      |     /    |
                      +----+-----+-------------> y
                     /     |    /
                    /      |   /
                   /       |  /
                  v x      * (x, y, 0) in Domain D

Evaluating Multivariable Functions

Evaluating a multivariable function requires substituting all input coordinates simultaneously into the expression. Order matters: $f(a, b)$ means $x = a$ and $y = b$ .

Engineering Applications

Thermodynamics & Heat Transfer: The Ideal Gas Law expresses pressure as a multivariable function of temperature and volume: $P(V, T) = \frac{nRT}{V}$ .
Civil & Structural Engineering: The deflection $\delta(x, y)$ across an elastic bridge deck slab subjected to vehicular loads.
Electrical & Computer Engineering: Power dissipation in a microchip processor $P(f, V) = C \cdot f \cdot V^2$ as a function of clock frequency $f$ and core voltage $V$ .
Aerodynamics: Drag force $F_D(\rho, v) = \frac{1}{2} C_D A \rho v^2$ acting on an aircraft fuselage as atmospheric density $\rho$ and flight velocity $v$ vary.

Example Problems

Example 1: Function Evaluation on a Thermodynamic Surface

Let $T(x, y) = 100 - 2x^2 - 3y^2$ describe the steady-state temperature (in $^{\circ}\text{C}$ ) on a thin metal cooling plate, where $x$ and $y$ are measured in centimeters from the plate center.

Evaluate the temperature at the center $(0,0)$ .
Evaluate the temperature at the edge coordinates $(2, -1)$ .
Determine the coordinates where $T(x, y) = 100$ .

Solution

Center $(0,0)$ : $T(0,0) = 100 - 2(0)^2 - 3(0)^2 = 100^{\circ}\text{C}$
Edge Point $(2, -1)$ : $T(2,-1) = 100 - 2(2)^2 - 3(-1)^2 = 100 - 2(4) - 3(1) = 100 - 8 - 3 = 89^{\circ}\text{C}$
Points where $T = 100$ : $100 - 2x^2 - 3y^2 = 100 \implies -2x^2 - 3y^2 = 0 \implies 2x^2 + 3y^2 = 0$ Since $x^2 \ge 0$ and $y^2 \ge 0$ , the only real solution is the origin $(0,0)$ .

Example 2: Three-Variable Function Evaluation

A scalar acoustic pressure field inside a reverberation chamber is modeled by:

p(x, y, z) = \frac{80}{1 + x^2 + 2y^2 + 3z^2}

Evaluate the pressure at the origin

(0, 0, 0)

and at the measurement point

(1, 2, 1)