In multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis. In physics, a basis is sometimes thought of as a set of reference axes.

What is the difference between contravariant and covariant?

In differential geometry, the components of a vector relative to a basis of the tangent bundle are covariant if they change with the same linear transformation as a change of basis. They are contravariant if they change by the inverse transformation.

What is the meaning of covariance and contravariance?

Covariance and contravariance are terms that refer to the ability to use a more derived type (more specific) or a less derived type (less specific) than originally specified. ... Means that you can use only the type originally specified. An invariant generic type parameter is neither covariant nor contravariant.

What is the meaning of contravariant?

Filters. (category theory, of a funct) Which reverses composition. adjective. (computing, programming) Using or relating to contravariance.

What do you mean by covariant tensor?

A covariant tensor, denoted with a lowered index (e.g., ) is a tensor having specific transformation properties. In general, these transformation properties differ from those of a contravariant tensor.

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29 related questions found

What are contravariant and covariant four vector?

If an object transforms in the same way as basis vectors, its a Covariant vector also called a 1-form. If it transforms as coordinates, and therefore in the opposite direction as basis vectors, it's a Contravariant vector, or just a vector.

Is force a contravariant?

In the language of Einstein-style index gymnastics, applied in a nonrelativistic context, this amounts to a statement that energy is a scalar, and displacement is a contravariant (upper-index) vector, so force should naturally be considered as a covariant (lower-index) vector.

Why is it called covariant derivative?

The name is motivated by the importance of changes of coordinate in physics: the covariant derivative transforms covariantly under a general coordinate transformation, that is, linearly via the Jacobian matrix of the transformation.

What does covariant mean in physics?

From Wikipedia, the free encyclopedia. In physics, the principle of covariance emphasizes the formulation of physical laws using only those physical quantities the measurements of which the observers in different frames of reference could unambiguously correlate.

What is covariance and Contravariance in C# and how is it used?

In C#, covariance and contravariance enable implicit reference conversion for array types, delegate types, and generic type arguments. Covariance preserves assignment compatibility and contravariance reverses it. ... string str = "test"; // An object of a more derived type is assigned to an object of a less derived type.

What is the covariance formula?

In statistics, the covariance formula helps to assess the relationship between two variables. It is essentially a measure of the variance between two variables. The covariance formula is expressed as, Covariance formula for population: Cov(X,Y)=∑(Xi−¯¯¯¯X)(Yi−¯¯¯¯Y)n C o v ( X , Y ) = ∑ ( X i − X ¯ ) ( Y i − Y ¯ ) n.

What is covariant in statistics?

Covariance is a measure of how much two random variables vary together. It's similar to variance, but where variance tells you how a single variable varies, co variance tells you how two variables vary together.

What are Covectors used for?

MEANING, USAGE & APPLICATIONS: Covectors are the essential structure for differential forms in differential topology/geometry and most of the important developments in those fields are formulated or use them in one way or another.

What does covariant mean in math?

(ˈkəʊˌvɛərɪənt) maths. n. (Mathematics) a variant that changes leaving interrelations with another variant (or variants) unchanged.

What is tensor with example?

A tensor field has a tensor corresponding to each point space. An example is the stress on a material, such as a construction beam in a bridge. Other examples of tensors include the strain tensor, the conductivity tensor, and the inertia tensor.

What does covariant form mean?

The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular, Maxwell's equations and the Lorentz force) in a form that is manifestly invariant under Lorentz transformations, in the formalism of special relativity using rectilinear inertial ...

Why is gradient covariant?

Gradient is covariant! ... The components of a vector contravariant because they transform in the inverse (i.e. contra) way of the vector basis. It is customary to denote these components with an upper index. So, if your coordinates are called q's, they are denoted qi.

What is contravariant derivative?

A "contravariant derivative operator" would probably be defined by , where is a torsion-free derivative operator that is compatible ( ) with a nondegenerate metric .

What is the difference between covariant derivative and Lie derivative?

Hopefully this illustrates the big differences between the two derivatives: the covariant derivative should be used to measure whether a tensor is parallel transported, while the Lie derivative measures whether a tensor is invariant under diffeomorphisms in the direction of the vector ξa.

What is Killing vector?

In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric.

Is covariant a basis vector?

The transformation that describes the new basis vectors as a linear combination of the old basis vectors is defined as a covariant transformation. Conventionally, indices identifying the basis vectors are placed as lower indices and so are all entities that transform in the same way.

What is covariant and invariant in Java?

Covariant would mean that a List<String> is a subtype of List<Object> , contravariant that a List<Object> is a subtype of List<String> and invariant that neither is a subtype of the other, i.e. List<String> and List<Object> are inconvertible types.

What is relationship between relativistic momentum and energy?

The energy–momentum relation is consistent with the familiar mass–energy relation in both its interpretations: E = mc² relates total energy E to the (total) relativistic mass m (alternatively denoted m_rel or m_tot ), while E₀ = m₀c² relates rest energy E₀ to (invariant) rest mass m₀.