What are covariant and contravariant tensors?

What are covariant and contravariant tensors?

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 contravariant metric tensor?

Also, the contravariant (covariant) forms of the metric tensor are expressed as the dot product of a pair of contravariant (covariant) basis vectors. Two vectors may be multiplied in the manner of a dot product, which produces a scalar, or in the manner of a cross product that produces another vector.

What is meant by 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. Generic type parameters support covariance and contravariance to provide greater flexibility in assigning and using generic types.

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 variant tensor?

A Variant Tensor can be a Tensor of any data type. Some examples of Variant Tensor are shown below: # Integer element a = 1 # Float element b = 2.0 # Tuple element with 2 components c = (1, 2) # Dict element with 3 components d = {“a”: (2, 2), “b”: 3} # Element containing a dataset e = tf.data.Dataset.from_element(10)

What are Covectors used for?

tldr covectors implement a “fix” to the dot product in differential geometry so that the result is invariant to changes of the local coordinate system.

What is covariance and contravariance in Java?

Covariance can be translated as “different in the same direction,” or with-different, whereas contravariance means “different in the opposite direction,” or against-different. Covariant and contravariant types are not the same, but there is a correlation between them.

Why is there no contravariant derivative?

It is a misnomer, but we are stuck with it. It is not the same “covariant” as that of a “covariant vector”, and therefore, there is no “contravariant derivative”.

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

Contravariance allows you to utilize a less derived type than originally specified, and covariance lets you use a more derived type. In a sense, the reason they were brought to the C# language is so you can extend arrays, delegate types and generic types with polymorphistic features.

What is the origin of the terms covariant and contravariant?

The terms covariant and contravariant were introduced by James Joseph Sylvester in 1851 in the context of associated algebraic forms theory. Tensors are objects in multilinear algebra that can have aspects of both covariance and contravariance.

Why do general tensors have contravariant indices and covariant indices?

The explanation in geometric terms is that a general tensor will have contravariant indices as well as covariant indices, because it has parts that live in the tangent bundle as well as the cotangent bundle . x μ {\\displaystyle x^ {\\mu }\\!}

What is covariance and contravariance in Computer Science?

Covariance and contravariance (computer science) By making type constructors covariant or contravariant instead of invariant, more programs will be accepted as well-typed. On the other hand, programmers often find contravariance unintuitive, and accurately tracking variance to avoid runtime type errors can lead to complex typing rules.

What is a covariant transformation law?

This is called a covariant transformation law, because the covector components transforms by the same matrix as the change of basis matrix. The components of a more general tensor transform by some combination of covariant and contravariant transformations, with one transformation law for each index.