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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. The notation grad f is also commonly used to ...
The gradient of a function is obtained by raising the index of the differential , whose components are given by: =; =; =, = = The divergence of a vector field with components is
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.
where here and below the Einstein notation is implied, so that the repeated index i is summed over. The gradient of a scalar function ƒ is the vector field grad f that may be defined through the inner product ⋅ , ⋅ {\displaystyle \langle \cdot ,\cdot \rangle } on the manifold, as
The Einstein tensor allows the Einstein field equations to be written in the concise form: + =, where is the cosmological constant and is the Einstein gravitational constant. From the explicit form of the Einstein tensor , the Einstein tensor is a nonlinear function of the metric tensor, but is linear in the second partial derivatives of the ...
For a stationary flow of an incompressible Newtonian fluid, these equations can be written in Einstein notation in Cartesian coordinates as: ¯ ¯ = ¯ + [¯ + (¯ + ¯) ′ ′ ¯]. The left hand side of this equation represents the change in mean momentum of a fluid element owing to the unsteadiness in the mean flow and the convection by the ...