Definition:Homogeneous

Disambiguation

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Homogeneous may refer to:

• Algebra:
  • Homogeneous quadratic equation: a quadratic equation in two variables in the form $a x^2 + h x y + b y^2 = 0$

• Analytic Geometry:
  • Homogeneous: a line or plane is homogeneous if it contains the origin.
  • Homogeneous Cartesian coordinates: such that $\tuple {x, y}$ is expressed as $\tuple {X, Y, Z}$ where $x = \dfrac X Z$ and $y = \dfrac Y Z$.

• Polynomial Theory
  • Homogeneous Polynomial

• Linear Algebra:
  • Homogeneous Linear Equations: a system of simultaneous equations which are all equal to zero.
  • Homogeneous function: a function $f: V \to W$ between two vector spaces over a field $F$ is homogeneous of degree $n$ if and only if $\map f {\alpha \mathbf v} = \alpha^n \map f {\mathbf v}$ for all nonzero $\mathbf v \in V$ and $\alpha \in F$.
    • Also see: homogeneous real function.

• Differential Equations:
  • Homogeneous differential equation: a first order ordinary differential equation of the form $\map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$, where both $M$ and $N$ are homogeneous functions.

• Model Theory:
  • Homogeneous: A concept in model theory.

• Metric Spaces:
  • Homogeneous: Another term for translation invariant.

• Physics:
  • Homogeneous: of a body, the same all the way through.

• Mathematical Models:
  • Homogeneous:

Linguistic Note

The word homogeneous comes from the Greek meaning of the same type.

Also see

• Definition:Homogenization