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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 ]
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.
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 ...
In Einstein's theory of general relativity, the Schwarzschild metric (also Schwarzschild vacuum or Schwarzschild solution), is a solution to the Einstein field equations which describes the gravitational field outside a spherical mass, on the assumption that the electric charge of the mass, the angular momentum of the mass, and the universal ...
The Einstein notation implies summation over i, since it appears as both an upper and lower index. The volume coefficient ρ is a function of position which depends on the coordinate system. In Cartesian, cylindrical and spherical coordinates, using the same conventions as before, we have ρ = 1 , ρ = r and ρ = r 2 sin θ , respectively.
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.
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 ...