Homogeneous Expression/Examples/Arbitrary Example 1

Example of Homogeneous Expression

The polynomial:

$\map E {x, y} = x^2 \map \sin {x / y} + y^2 \map \cos {x / y}$

is a homogeneous expression.

This is demonstrated by replacing $x$ by $k x$ and $y$ by $k y$:

\(\ds \map E {k x, k y}\)

\(=\)

\(\ds k^2 x^2 \map \sin {k x / k y} + k^2 y^2 \map \cos {k x / k y}\)

\(\ds \)

\(=\)

\(\ds k^2 \paren {x^2 \map \sin {x / y} + y^2 \map \cos {x / y} }\)

\(\ds \)

\(=\)

\(\ds k^2 \map E {x, y}\)

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): homogeneous
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): homogeneous