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.As part of mathematics it is a notational subset of Ricci calculus; however, it is often used in physics applications that do not distinguish between tangent and cotangent spaces.[1] According to this convention, when an index variable appears twice in a single term and is not otherwise defined (see Free and bound variables), it implies summation of that term over all the values of the index.That is, in this context x2 should be understood as the second component of x rather than the square of x (this can occasionally lead to ambiguity).The upper index position in xi is because, typically, an index occurs once in an upper (superscript) and once in a lower (subscript) position in a term (see § Application below).In general relativity, a common convention is that In general, indices can range over any indexing set, including an infinite set.This should not be confused with a typographically similar convention used to distinguish between tensor index notation and the closely related but distinct basis-independent abstract index notation.It is also called a dummy index since any symbol can replace "i " without changing the meaning of the expression (provided that it does not collide with other index symbols in the same term).Einstein notation can be applied in slightly different ways.Typically, each index occurs once in an upper (superscript) and once in a lower (subscript) position in a term; however, the convention can be applied more generally to any repeated indices within a term.In terms of covariance and contravariance of vectors, They transform contravariantly or covariantly, respectively, with respect to change of basis.In recognition of this fact, the following notation uses the same symbol both for a vector or covector and its components, as in:are each column vectors, and the covector basis elements(See also § Abstract description; duality, below and the examples) In the presence of a non-degenerate form (an isomorphism V → V∗, for instance a Riemannian metric or Minkowski metric), one can raise and lower indices.However, if one changes coordinates, the way that coefficients change depends on the variance of the object, and one cannot ignore the distinction; see Covariance and contravariance of vectors.In the above example, vectors are represented as n × 1 matrices (column vectors), while covectors are represented as 1 × n matrices (row covectors).When using the column vector convention: The virtue of Einstein notation is that it represents the invariant quantities with a simple notation.In physics, a scalar is invariant under transformations of basis.This led Einstein to propose the convention that repeated indices imply the summation is to be done.This is designed to guarantee that the linear function associated with the covector, the sum above, is the same no matter what the basis is.The value of the Einstein convention is that it applies to other vector spaces built from V using the tensor product and duality.V *, the dual of V, has a basis e1, e2, ..., en which obeys the rulethe row/column coordinates on a matrix correspond to the upper/lower indices on the tensor product.In Einstein notation, the usual element referenceThe inner product of two vectors is the sum of the products of their corresponding components, with the indices of one vector lowered (see #Raising and lowering indices):In three dimensions, the cross product of two vectors with respect to a positively oriented orthonormal basis, meaning thatThe product of a matrix Aij with a column vector vj is:The matrix product of two matrices Aij and Bjk is:For a square matrix Aij, the trace is the sum of the diagonal elements, hence the sum over a common index Aii.For example, taking the tensor Tαβ, one can lower an index: