# Definition:Homogeneous

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## Disambiguation

This page lists articles associated with the same title. If an internal link led you here, you may wish to change the link to point directly to the intended article.

**Homogeneous** may refer to:

- Algebra:
- Homogeneous expression: an algebraic expression in which the variables can be replaced throughout by the product of that variable with a given non-zero constant, and the constant can be extracted as a factor of the resulting expression
- Homogeneous equation: a homogeneous expression equated to zero
- 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 $y = \dfrac Y Z$.

- Homogeneous: a line or plane is

- 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.

- Integral Equations:
- Homogeneous integral equation of the second kind: an integral equation of the form $\map g x = \lambda \ds \int_{\map a x}^{\map b x} \map K {x, y} \map g y \rd x$

- 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.

## Linguistic Note

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