Diffeomorphisms | Differential geometry

General covariance

In theoretical physics, general covariance, also known as diffeomorphism covariance or general invariance, consists of the invariance of the form of physical laws under arbitrary differentiable coordinate transformations. The essential idea is that coordinates do not exist a priori in nature, but are only artifices used in describing nature, and hence should play no role in the formulation of fundamental physical laws. While this concept is exhibited by general relativity, which describes the dynamics of spacetime, one should not expect it to hold in less fundamental theories. For matter fields taken to exist independently of the background, it is almost never the case that their equations of motion will take the same form in curved space that they do in flat space. (Wikipedia).

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Covariance (1 of 17) What is Covariance? in Relation to Variance and Correlation

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From playlist COVARIANCE AND VARIANCE

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From playlist Summer of Math Exposition 2 videos

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From playlist Correlation

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Covariance (8 of 17) What is the Correlation Coefficient?

Visit http://ilectureonline.com for more math and science lectures! To donate:a http://www.ilectureonline.com/donate https://www.patreon.com/user?u=3236071 We will learn what is and how to find the correlation coefficient of 2 data sets and see how it corresponds to the graph of the data

From playlist COVARIANCE AND VARIANCE

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Covariance (6 of 17) Example of the Covariance Matrix - EX 1

Visit http://ilectureonline.com for more math and science lectures! To donate:a http://www.ilectureonline.com/donate https://www.patreon.com/user?u=3236071 We will find the covariance matrix of 2 data sets. Example 1 Next video in this series can be seen at: https://youtu.be/9DscP6F5CGs

From playlist COVARIANCE AND VARIANCE

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From playlist COVARIANCE AND VARIANCE

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Covariance (14 of 17) Covariance Matrix "Normalized" - Correlation Coefficient

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From playlist COVARIANCE AND VARIANCE

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Covariance (12 of 17) Covariance Matrix wth 3 Data Sets and Correlation Coefficients

Visit http://ilectureonline.com for more math and science lectures! To donate:a http://www.ilectureonline.com/donate https://www.patreon.com/user?u=3236071 We will find the correlation coefficients of the 3 data sets form the previous 2 videos. Next video in this series can be seen at:

From playlist COVARIANCE AND VARIANCE

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Neuroscience source separation 2a: Spatial separation

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From playlist Neuroscience source separation (3-part lecture series)

Related pages

Lorentz covariance | Covariant derivative | Lie group | Metric tensor | Gauge covariant derivative | Differential geometry | Covariance and contravariance of vectors | Tensor field | Diffeomorphism | Geodesics in general relativity | Invariant (physics) | Derivative | Symmetry | Albert Einstein | General covariant transformations | Group (mathematics) | Minkowski space