Calculus I:
functions of several variables
Definitions

Definitions

Let's see the simplest example of two-variable functions. It is the area of a rectangle: A = L x l. L and l being positive numbers, we represent this according to the following manner:

f: R⁺ x R⁺ → R⁺

(L, l) → L x l

R⁺ x R⁺ = R⁺² is called the domain of definition of the function f.

In a general way we can have n variables where n denotes an integer.

Let n be an integer and D a subset of Rn. A function f of n variables is a process that for any n-tuple (x1, ..., xn) of D , it is associated a unique real number.

This is noted in the following way:

f: D → R
(x1, ..., xn) → f (x1, ..., xn)
D is the domain of definition of f.

Note: The notation (x1, ..., xn) is here to show that we have n variables. In practice, when we only have two variables, we write x and y rather than x1 and x2.

For example, the following function gives the distance of a coordinate point (x, y) from the origin of the plan.

f: R² → R
(x, y) → √(x² + y²)

f is a function of two variables, R² is its domain of definition.

Here is an example of a function of three variables: (x; y; z).

g: R x R x R^* → R
(x, y, z) → (z cos(x) + 3x⁵ - 3)/y

g is a function of three variables, R x R x R^* is his area of definition.

Exercise 1.

The following formula defines a function of 2 variables:

f (x, y) = sin (x) + ln(y)

1. Give the image of (0, e).
2. Give the largest domaine of definition possible for f.

Solution:

1. f (0, e) = sin (0) + ln(e) = 0 + 1 = 1.
The image of (0, e) by f is 1.

2. sin (x) exists for all x belongs R. So y belongs R.
For ln (y) to exist x must be positive: x > 0. So x belongs R⁺.

So the biggest possible domain of definition for f is: R x R⁺ .