Definition:Real Function/Two Variables

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Definition

Let $S, T \subseteq \R$ be subsets of the set of real numbers $\R$.

Let $f: S \times T \to \R$ be a mapping.

Then $f$ is defined as a (real) function of two (independent) variables.


The expression:

$z = \map f {x, y}$

means:

(The dependent variable) $z$ is a function of (the independent variables) $x$ and $y$.


Substitution for $y$

Let $f: S \times T \to \R$ be a (real) function of two variables:

$z = \map f {x, y}$

Then:

$\map f {x, a}$

means the real function of $x$ obtained by replacing the independent variable $y$ with $a$.

In this context, $a$ can be:

a real constant such that $a \in T$
a real function $\map g x$ whose range is a subset of $T$.


Examples

Volume of Right Circular Cylinder

Let $C$ denote be a right circular cylinder.


Then the volume $V$ of $C$ is a function of:

the radius $r$ of the base of $C$
the height $h$ of $C$

such that:

$V = \pi r^2 h$


Example: $y \sqrt {1 - x^2}$

Let $z$ denote the function defined as:

$z = y \sqrt {1 - x^2}$

The domain of $z$ is:

$\Dom z = \closedint {-1} 1 \times \R$


Example: $\dfrac {\sqrt {1 - y^2} } {\sqrt {1 - x^2} }$

Let $z$ denote the function defined as:

$z = \dfrac {\sqrt {1 - y^2} } {\sqrt {1 - x^2} }$

The domain of $z$ is:

$\Dom z = \openint {-1} 1 \times \closedint {-1} 1$


Sources

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