Functions of Several Variables

Most engineering quantities depend on several variables: temperature T(x, y, z, t), pressure p(V, T), stress σ(x, y). Learn functions of two and three variables, domains, surfaces, level curves and contour maps, level surfaces, and scalar and vector physical fields.

A weather map, a heat map of a CPU, or a stress plot from FEA software are all visualisations of multivariable functions. Understanding them is the first step toward partial derivatives, gradients and multiple integrals.

An analogy

A contour map of a mountain is a 2D picture of a 3D function: each contour line joins points of equal height.

Surfaces and domains

z = f(x, y) is a surface above the xy-plane. Domains are regions: ln(x + y) needs x + y > 0; √(4 − x² − y²) needs the disc x² + y² ≤ 4.

Contours and level surfaces

Isotherms, isobars and equipotential lines are level curves. Closely spaced contours mean steep change. For three variables, level surfaces (e.g. equal-temperature shells) play the same role.

Scalar and vector fields

Scalar fields assign a number to each point (temperature, pressure, potential). Vector fields assign a vector (velocity, force, electric field). Both are central to vector calculus.

Limits and continuity in 2D

A 2D limit must be the same along every path. xy/(x² + y²) approaches different values along y = x and y = 0, so it has no limit at the origin.

Formulas

• Ideal gas: p(V, T) = nRT / V
• Paraboloid: z = x² + y² (level curves: circles x² + y² = c)
• Distance field: r(x, y, z) = √(x² + y² + z²) (level surfaces: spheres)

Worked example: Domain

Given: f(x, y) = √(4 − x² − y²)

Find: Domain and range.

• 4 − x² − y² ≥ 0 → x² + y² ≤ 4
• Range: 0 ≤ f ≤ 2

Answer: Closed disc of radius 2; range [0, 2] (a hemisphere)

Worked example: Ideal gas

Given: n = 1 mol, R = 8.314, V = 0.0224 m³, T = 273 K

Find: p.

• p = 8.314 × 273/0.0224 = 101 300 Pa

Answer: ≈ 1 atm

Common mistakes

• Wrong: Checking a 2D limit along one path only. — Right: Different paths can give different values.
• Wrong: Reading contour spacing as value. — Right: Spacing shows gradient; labels show value.

Applications

• Thermal maps: Temperature fields in electronics and buildings.
• Terrain & surveying: Contour maps for earthworks.
• FEA results: Stress and displacement fields.

Key takeaways

• f(x, y) is a surface.
• Level curves/surfaces show equal values.
• Scalar vs vector fields.
• 2D limits must agree along all paths.

Going further

• Visualisation: Surface, contour and quiver plots in Python/MATLAB.
• Tensor fields: Stress as a field of tensors (M16).
• Multivariable limits: Polar substitution tests.