covariant

Thân thiện

A mathematician writes a covariant equation on the blackboard.

Definition

1. Adjective:
   • Changing in a way that maintains a consistent mathematical relationship with another variable or set of variables: In mathematics and physics, 'covariant' describes a quantity whose transformation under a change of coordinates or conditions is directly linked to the transformation of another quantity, preserving the form of their relationship.

Usage

• General Use: The term is primarily used in technical fields like mathematics, physics, and statistics. It describes how entities change together under specific operations.
  • In tensor calculus, a covariant vector transforms in a specific way under coordinate changes.
  • The equations are written in a covariant form, meaning they hold true in all frames of reference.

Advanced Usage

• Covariant Derivative: In differential geometry, this is a derivative that accounts for how the basis vectors of a space change, allowing for differentiation on curved surfaces.
  • The covariant derivative generalizes the concept of a directional derivative to manifolds.
• Covariant Functor: In category theory, a functor that preserves the direction of morphisms (arrows) between categories.
  • A forgetful functor from the category of groups to the category of sets is a covariant functor.

Variants and Related Words

• Covariance (n): The property of being covariant; also a statistical measure of the joint variability of two random variables.
  • The covariance of the two stocks indicated they often moved together.
• Contravariant (adj): The complementary concept, where a quantity transforms in an inverse manner relative to the basis vectors.
  • While a vector can be covariant, its dual vector is contravariant.

Synonyms

• Corresponding: Changing in a related or parallel manner.
• Coordinated: Changing together in a systematic way.

Related Phrases

• Covariant transformation: The specific rule or operation that defines how a covariant object changes.
  • Understanding covariant transformation is key to general relativity.
• Covariant formulation: A way of expressing physical laws so they are valid in all coordinate systems.
  • Maxwell's equations have an elegant covariant formulation in special relativity.

Related Idioms

(This term is highly technical and does not have common idiomatic expressions in everyday language.)

Học thuật

Adjective

1. changing so that interrelations with another variable quantity or set of quantities remain unchanged