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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.
For example, summation of [1, 2, 4, 2] is denoted 1 + 2 + 4 + 2, and results in 9, that is, 1 + 2 + 4 + 2 = 9. Because addition is associative and commutative, there is no need for parentheses, and the result is the same irrespective of the order of the summands. Summation of a sequence of only one summand results in the summand itself.
In index-free tensor notation, the Levi-Civita symbol is replaced by the concept of the Hodge dual. [citation needed] Summation symbols can be eliminated by using Einstein notation, where an index repeated between two or more terms indicates summation over that index. For example,
Summation notation may refer to: Capital-sigma notation, mathematical symbol for summation; Einstein notation, summation over like-subscripted indices
Summation methods include Cesàro summation, generalized Cesàro (,) summation, Abel summation, and Borel summation, in order of applicability to increasingly divergent series. These methods are all based on sequence transformations of the original series of terms or of its sequence of partial sums.
1. Internal direct sum: if E and F are abelian subgroups of an abelian group V, notation = means that V is the direct sum of E and F; that is, every element of V can be written in a unique way as the sum of an element of E and an element of F.
In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums. It is also called Abel's lemma or Abel transformation , named after Niels Henrik Abel who introduced it in 1826.
In addition to compactly supported functions and integrable functions, functions that have sufficiently rapid decay at infinity can also be convolved. An important feature of the convolution is that if f and g both decay rapidly, then f∗g also decays rapidly. In particular, if f and g are rapidly decreasing functions, then so is the ...