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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.
The utility of the Feynman subscript notation lies in its use in the derivation of vector and tensor derivative identities, as in the following example which uses the algebraic identity C⋅(A×B) = (C×A)⋅B:
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 ...
The gradient of the function f(x,y) = −(cos 2 x + cos 2 y) 2 depicted as a projected vector field on the bottom plane. The gradient (or gradient vector field) of a scalar function f(x 1, x 2, x 3, …, x n) is denoted ∇f or ∇ → f where ∇ denotes the vector differential operator, del. The notation grad f is also commonly used to ...
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 ...
where the Feynman subscript notation is used, which means the subscripted gradient operates only on the factor . Lamb in his famous classical book Hydrodynamics (1895), still in print, used this identity to change the convective term of the flow velocity in rotational form: [ 14 ]
For the special case where the tensor product operation is a contraction of one index and the gradient operation is a divergence, and both and are second order tensors, we have = ():. In index notation, ∫ Ω F i j G p j , p d Ω = ∫ Γ n p F i j G p j d Γ − ∫ Ω G p j F i j , p d Ω . {\displaystyle \int _{\Omega }F_{ij}\,G_{pj,p}\,d ...
An affine connection defines a notion of development of curves. Intuitively, development captures the notion that if x t is a curve in M, then the affine tangent space at x 0 may be rolled along the curve. As it does so, the marked point of contact between the tangent space and the manifold traces out a curve C t in this affine space: the ...