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One difference between the Gaussian and SI systems is in the factor 4π in various formulas that relate the quantities that they define. With SI electromagnetic units, called rationalized, [3] [4] Maxwell's equations have no explicit factors of 4π in the formulae, whereas the inverse-square force laws – Coulomb's law and the Biot–Savart law – do have a factor of 4π attached to the r 2.
Consequently, Gaussian functions are also associated with the vacuum state in quantum field theory. Gaussian beams are used in optical systems, microwave systems and lasers. In scale space representation, Gaussian functions are used as smoothing kernels for generating multi-scale representations in computer vision and image processing.
The electromagnetic stress–energy tensor in the International System of Quantities (ISQ), which underlies the SI, is [1] = [], where is the electromagnetic tensor and where is the Minkowski metric tensor of metric signature (− + + +) and the Einstein summation convention over repeated indices is used.
The Gaussian quadrature chooses more suitable points instead, so even a linear function approximates the function better (the black dashed line). As the integrand is the polynomial of degree 3 (y(x) = 7x 3 – 8x 2 – 3x + 3), the 2-point Gaussian quadrature rule even returns an exact result.
All these extensions are also called normal or Gaussian laws, so a certain ambiguity in names exists. The multivariate normal distribution describes the Gaussian law in the k-dimensional Euclidean space. A vector X ∈ R k is multivariate-normally distributed if any linear combination of its components Σ k j=1 a j X j has a (univariate) normal ...
In terms of the Gaussian base units, it is 1 statC = 1 dyn 1/2 ⋅cm = 1 cm 3/2 ⋅g 1/2 ⋅s −1 . That is, it is defined so that the proportionality constant in Coulomb's law using CGS-ESU quantities is a dimensionless quantity equal to 1.
Gaussian measures with mean = are known as centered Gaussian measures. The Dirac measure δ μ {\displaystyle \delta _{\mu }} is the weak limit of γ μ , σ 2 n {\displaystyle \gamma _{\mu ,\sigma ^{2}}^{n}} as σ → 0 {\displaystyle \sigma \to 0} , and is considered to be a degenerate Gaussian measure ; in contrast, Gaussian measures with ...
Common integrals in quantum field theory are all variations and generalizations of Gaussian integrals to the complex plane and to multiple dimensions. [1]: 13–15 Other integrals can be approximated by versions of the Gaussian integral. Fourier integrals are also considered.