Definition:Homogeneous
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Homogeneous may refer to:
- Homogeneous (Analytic Geometry): a line or plane is homogeneous if it contains the origin.
- Homogeneous Linear Equations: a system of simultaneous equations which are all equal to zero.
- Homogeneous polynomial: a polynomial whose mononomials with nonzero coefficients all have the same total degree.
- Homogeneous function: a function $f: V \to W$ between two vector spaces over a field $F$ is homogeneous of degree $n$ if $f \left({\alpha \mathbf v}\right) = \alpha^n f \left({\mathbf v}\right)$ for all nonzero $\mathbf v \in V$ and $\alpha \in F$.
- Homogeneous differential equation: a first order ordinary differential equation of the form $\displaystyle M \left({x, y}\right) + N \left({x, y}\right) \frac {\mathrm d y} {\mathrm d x} = 0$, where both $M$ and $N$ are homogeneous functions.
- Homogeneous (Model Theory): A concept in model theory.
- Homogeneous (Metric Spaces): Another term for translation invariance.
- Homogeneous (Physics): of a body, the same all the way through.
Linguistic Note
The word homogeneous comes from the Greek meaning of the same type.
