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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 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 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 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.
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 .
We'll cover exactly how to play Strands, hints for today's spangram and all of the answers for Strands #320 on Friday, January 17. Related: 16 Games Like Wordle To Give You Your Word Game Fix More ...
e 1, e 2, e 3 to the coordinate curves (left), dual basis, covector basis, or reciprocal basis e 1, e 2, e 3 to coordinate surfaces (right), in 3-d general curvilinear coordinates (q 1, q 2, q 3), a tuple of numbers to define a point in a position space. Note the basis and cobasis coincide only when the basis is orthonormal. [1] [specify]