In basic (that is to say, one-dimensional
, 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
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 Rm
, the ith component function of f, denoted fi
, 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.