Functions of several variables

In first-year calculus, functions had one input: the deflection of a beam at position \(x\), the velocity at time \(t\). Real engineering quantities usually depend on several inputs at once. The temperature in a steel plate depends on two coordinates; the pressure of a gas depends on its volume and temperature; the stress in a dam depends on three coordinates; the wind chill you feel depends on air temperature and wind speed. Multivariable calculus extends derivatives and integrals to such functions, and it is the mathematical language of fluid mechanics, heat transfer, electromagnetics, solid mechanics and optimization.

This first chapter builds intuition before calculus: how to describe functions of two and three variables, picture them as surfaces and contour maps, read engineering contour plots, and understand what limits and continuity mean when you can approach a point from infinitely many directions.

Functions of two variables

A function of two variables assigns one number \(z = f(x, y)\) to each point \((x, y)\) in a set called the domain. The set of output values is the range. As in single-variable calculus, the domain is either stated or taken as every input for which the formula makes sense.

Engineering functions often come from formulas fitted to data or derived from physics, and their domains carry physical meaning: absolute temperature must be positive, a length cannot be negative, a fitted formula may only be valid over the range of the tests behind it.

Graphs and surfaces

The graph of \(f(x, y)\) is the set of points \((x, y, z)\) with \(z = f(x, y)\): a surface in three dimensions, sitting above (or below) the domain in the xy-plane. You can picture the function as a landscape, with \(z\) as the height of the ground above each map point.

Some surfaces appear constantly and are worth recognizing from their equations:

• A plane: \(z = ax + by + c\).
• A paraboloid (bowl or dome): \(z = x^2 + y^2\) opens upward; \(z = 100 - x^2 - 2y^2\) is an elliptic dome opening downward.
• A saddle (hyperbolic paraboloid): \(z = x^2 - y^2\) curves up along the x-axis and down along the y-axis. Saddle shapes appear in roof structures and in the analysis of critical points in Lesson 2.
• A cone: \(z = \sqrt{x^2 + y^2}\).
• A hemisphere: \(z = \sqrt{r^2 - x^2 - y^2}\), the function in the domain example above.

Level curves and contour maps

Drawing surfaces is hard and reading values off them is harder. Engineers far more often use contour maps. A level curve (or contour) of \(f\) is the set of points where \(f(x, y) = k\) for a constant \(k\). A contour map shows several level curves, usually at equal intervals of \(k\), on the xy-plane.

You use contour maps already: topographic maps show elevation contours, weather maps show isobars (equal pressure) and isotherms (equal temperature). In engineering they show stress in a component, temperature in a heat sink, groundwater head under a dam, and noise levels around an airport.

Contour spacing carries information. Where contours are close together the function changes rapidly; where they are far apart it changes slowly. On a topographic map, tight contours mean a steep slope. In the plate, the isotherms crowd together near the edges, so heat flows fastest there. In Lesson 2, the gradient will make this precise: it points straight across the contours, and its size is inversely related to their spacing.

Functions of three or more variables

A function \(w = f(x, y, z)\) assigns a number to each point in space: the temperature in a room, the concentration of a pollutant in an aquifer, the electric potential around a charge. Its graph would need four dimensions, so instead we picture level surfaces, where \(f(x, y, z) = k\).

Engineering models routinely have many more variables. The ideal gas law \(P = nRT/V\) has three; the cost of a pipeline depends on diameter, wall thickness, length, material and pump power. The calculus you learn for two and three variables extends directly to any number, which is why the methods of this course underpin optimization software that handles thousands.

Limits and continuity

The limit \(\lim_{(x,y)\to(a,b)} f(x, y) = L\) means that \(f(x, y)\) gets arbitrarily close to \(L\) as \((x, y)\) gets close to \((a, b)\), by any route. This last phrase is the new difficulty. On a line you can approach a point only from the left or the right. In the plane you can approach along any line, any curve, or a spiral, and the limit exists only if every route gives the same value.

Most functions built from polynomials, roots, exponentials and trigonometric functions are continuous wherever they are defined, so their limits are found by substitution. The interesting cases are points where the formula breaks down, typically where a denominator is zero.

The converse is not true: checking every straight line is not enough, because a function can agree along all lines and still fail along a curve such as \(y = x^2\). To prove that a limit does exist, bound the function. Polar coordinates are often the easiest way.

A function is continuous at a point if its limit there equals its value. Continuity matters in engineering because many results, including the existence of a maximum on a closed region (used for optimization in Lesson 2), depend on it, and because numerical methods behave badly near discontinuities such as cracks in a solid or shocks in a gas flow.