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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 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 ...
For example, the gradient of the function (,,) = + is (,,) = + (). or (,,) = []. In some applications it is customary to represent the gradient as a row vector or column vector of its components in a rectangular coordinate system; this article follows the convention of the gradient being a column vector, while the derivative is a row ...
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 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 ]
When studying and formulating Albert Einstein's theory of general relativity, various mathematical structures and techniques are utilized. The main tools used in this geometrical theory of gravitation are tensor fields defined on a Lorentzian manifold representing spacetime. This article is a general description of the mathematics of general ...
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 .
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 ...