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  2. Einstein notation - Wikipedia

    en.wikipedia.org/wiki/Einstein_notation

    In mathematics, especially the usage of linear algebra in mathematical physics and differential geometry, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving brevity.

  3. Covariance and contravariance of vectors - Wikipedia

    en.wikipedia.org/wiki/Covariance_and_contra...

    In Einstein notation (implicit summation over repeated index), contravariant components are denoted with upper indices as in = A covector or cotangent vector has components that co-vary with a change of basis in the corresponding (initial) vector space. That is, the components must be transformed by the same matrix as the change of basis matrix ...

  4. Vector calculus identities - Wikipedia

    en.wikipedia.org/wiki/Vector_calculus_identities

    In Einstein notation, ... Less general but similar is the Hestenes overdot notation in geometric algebra. [3] The above identity is then expressed as: ...

  5. Einstein field equations - Wikipedia

    en.wikipedia.org/wiki/Einstein_field_equations

    The Einstein field equations (EFE) may be written in the form: [5] [1] + = EFE on a wall in Leiden, Netherlands. where is the Einstein tensor, is the metric tensor, is the stress–energy tensor, is the cosmological constant and is the Einstein gravitational constant.

  6. Mathematics of general relativity - Wikipedia

    en.wikipedia.org/wiki/Mathematics_of_general...

    In abstract index notation, the EFE reads as follows: + = where is the Einstein tensor, is the cosmological constant, is the metric tensor, is the speed of light in vacuum and is the gravitational constant, which comes from Newton's law of universal gravitation.

  7. Introduction to the mathematics of general relativity - Wikipedia

    en.wikipedia.org/wiki/Introduction_to_the...

    The Einstein tensor G is a rank-2 tensor defined over pseudo-Riemannian manifolds. In index-free notation it is defined as =, where R is the Ricci tensor, g is the metric tensor and R is the scalar curvature. It is used in the Einstein field equations.

  8. Ricci calculus - Wikipedia

    en.wikipedia.org/wiki/Ricci_calculus

    For compactness and convenience, the Ricci calculus incorporates Einstein notation, which implies summation over indices repeated within a term and universal quantification over free indices. Expressions in the notation of the Ricci calculus may generally be interpreted as a set of simultaneous equations relating the components as functions ...

  9. Christoffel symbols - Wikipedia

    en.wikipedia.org/wiki/Christoffel_symbols

    where (g jk) is the inverse of the matrix (g jk), defined as (using the Kronecker delta, and Einstein notation for summation) g ji g ik = δ j k. Although the Christoffel symbols are written in the same notation as tensors with index notation, they do not transform like tensors under a change of coordinates.