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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.
In Feynman subscript notation, = + where the notation ∇ B means the subscripted gradient operates on only the factor B. [ 1 ] [ 2 ] Less general but similar is the Hestenes overdot notation in geometric algebra . [ 3 ]
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 ...
The free Euler equations are conservative, in the sense they are equivalent to a conservation equation: + =, or simply in Einstein notation: + =, where the conservation quantity in this case is a vector, and is a flux matrix. This can be simply proved.
Covariant vectors, on the other hand, have units of one-over-distance (as in a gradient) and transform in the same way as the coordinate system. For example, in changing from meters to millimeters, the coordinate units become smaller and the number measuring a gradient will also become smaller: 1 Kelvin per m becomes 0.001 Kelvin per mm.
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 ...
If ,, are the contravariant basis vectors in a curvilinear coordinate system, with coordinates of points denoted by (,,), then the gradient of the tensor field is given by (see [3] for a proof.) = From this definition we have the following relations for the gradients of a scalar field ϕ {\displaystyle \phi } , a vector field v , and a second ...