stylistic:tone, structural:scholar_edition
Examples of real-valued functions of several real variables

Examples in continuum mechanics include the local mass density ρ of a mass distribution, a scalar field which depends on the spatial position coordinates (here Cartesian to exemplify), r = (x, y, z), and time t:

Similarly for electric charge density for electrically charged objects, and numerous other scalar potential fields.

Another example is the velocity field, a vector field, which has components of velocity v = (v, v, v) that are each multivariable functions of spatial coordinates and time similarly:

Similarly for other physical vector fields such as electric fields and magnetic fields, and vector potential fields.

Another important example is the equation of state in thermodynamics, an equation relating pressure P, temperature T, and volume V of a fluid, in general it has an implicit form:

The simplest example is the ideal gas law:

where n is the number of moles, constant for a fixed amount of substance, and R the gas constant. Much more complicated equations of state have been empirically derived, but they all have the above implicit form.

Real-valued functions of several real variables appear pervasively in economics. In the underpinnings of consumer theory, utility is expressed as a function of the amounts of various goods consumed, each amount being an argument of the utility function. The result of maximizing utility is a set of demand functions, each expressing the amount demanded of a particular good as a function of the prices of the various goods and of income or wealth. In producer theory, a firm is usually assumed to maximize profit as a function of the quantities of various goods produced and of the quantities of various factors of production employed. The result of the optimization is a set of demand functions for the various factors of production and a set of supply functions for the various products; each of these functions has as its arguments the prices of the goods and of the factors of production.