Real Function of Two Variables/Examples

Examples of Real Functions of Two Variables

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:

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

Example: $x + y$

Let $z$ denote the function defined as:

$z = x + y$

The domain of $z$ is:

$\Dom z = \R \times \R$

Example: $\sqrt {x^2 + y^2 - 25}$

Let $z$ denote the function defined as:

$z = \sqrt {x^2 + y^2 - 25}$

The domain of $z$ is:

$\Dom z = C$

where $C$ consists of the set of points outside and on the circumference of the circle of radius $5$ whose center is at $\tuple {0, 0}$ in the Cartesian plane.

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

Let $z$ denote the function defined as:

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

The domain of $z$ is:

$\Dom z = \tuple {0, 0}$

That is, just one single point, the origin of the Cartesian plane

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

Let $z$ denote the function defined as:

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

The domain of $z$ is:

$\Dom z = \O$

That is, there are no points of the Cartesian plane for which $z$ is defined.

Arbitrary Example

The real-valued function $f: \R^2 \to \R$:

$\map f {x_1, x_2} = \sin x_1 + x_1 \cos x_2$

is an example of a real function of two wariables.