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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.
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.
In abstract index notation, the EFE reads as follows: + = where is the Einstein tensor, is the cosmological constant, is the metric tensor, is the speed of light in vacuum and is the gravitational constant, which comes from Newton's law of universal gravitation.
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 ...
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
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 ...
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 .