In basic (that is to say,
one-dimensional)
calculus, a
function is a rule that assigns to every
real number in its
domain some other real number. It is under this framework that certain essential terms (
continuous and
differentiable functions, for example) are defined. When moving from one-variable calculus to
multivariable calculus, we wish to preserve as many of these concepts as possible. Unfortunately, many of these concepts are defined in a way that assumes implicitly that the
output of a function will be a single real number. In order to generalize these concepts, we can introduce the concept of a component function. If f is a function that maps a set of points A in
Rn to points in R
m, the ith component function of f, denoted f
i, is defined as follows:
For all points x in A, if f(x) = (a1, ... , am), then fi(x) = ai.
To put it another way, f(x) = (f1(x), ... , fm(x)) for all points x in A. Therefore, the range of these component functions lies in R.
Example: If f(x,y) = (2x + y, 3xy), f1(x,y) = 2x + y, which for specific values of x and y yields a number in R.
The jth partial derivative of the ith component function of f is denoted Djfi. This term is useful because it can be shown that if f is differentiable, each Djfi exists, and while the converse is not true, a slightly stronger condition is sufficient to guarantee that a function is continuously differentiable.