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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 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
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 ...
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.
Using these properties, the Navier–Stokes equations of motion, expressed in tensor notation, are (for an incompressible Newtonian fluid): = + = + where is a vector representing external forces. Next, each instantaneous quantity can be split into time-averaged and fluctuating components, and the resulting equation time-averaged, [ b ] to yield:
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In differential geometry, the four-gradient (or 4-gradient) is the four-vector analogue of the gradient from vector calculus. In special relativity and in quantum mechanics , the four-gradient is used to define the properties and relations between the various physical four-vectors and tensors .