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:
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.