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Examples ofcovariant derivative
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Connections (gauge connection) define this principal bundle, yielding acovariantderivativein each associated vector bundle.
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This example is from Wikipedia and may be reused under a CC BY-SA license.The operator can be extended to operate on tensors as the divergence of thecovariantderivative.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Even to formulate such equations requires a fresh notion, thecovariantderivative.
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This example is from Wikipedia and may be reused under a CC BY-SA license.This regulator is gauge invariant due to the auxiliary particles being minimally coupled to the photon field through the gaugecovariantderivative.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Thecovariantderivativeof a tensor field is presented as an extension of the same concept.
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This example is from Wikipedia and may be reused under a CC BY-SA license.In words: thecovariantderivativeis the usual derivative along the coordinates with correction terms which tell how the coordinates change.
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This example is from Wikipedia and may be reused under a CC BY-SA license.In addition to its interaction with other objects via thecovariantderivative, the gauge field typically contributes energy in the form of a self-energy term.
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This example is from Wikipedia and may be reused under a CC BY-SA license.The connection associated with thiscovariantderivativeis completely determined by the tetrad.
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This example is from Wikipedia and may be reused under a CC BY-SA license.In curved space-time, we must take thecovariantderivative.
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This example is from Wikipedia and may be reused under a CC BY-SA license.The equivalence of the more traditional general relativistic formulation using thecovariantderivative with the differential form formulation can be seen as follows.
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This example is from Wikipedia and may be reused under a CC BY-SA license.This is the tactic taken by thecovariantderivativeapproach to connections: good behavior is equated with covariance.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Thecovariantderivativecan in turn be recovered from parallel transport.
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This example is from Wikipedia and may be reused under a CC BY-SA license.The equivalence principle is not assumed, but instead follows from the fact that the gaugecovariantderivativeis minimally coupled.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Thecovariantderivativeis convenient however because it commutes with raising and lowering indices.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Thecovariantderivativeof a tensor field along a vector field v is again a tensor field of the same type.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Often a notation is used in which thecovariantderivativeis given with a semicolon, while a normal partial derivative is indicated by a comma.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Furthermore, many of the features of thecovariantderivativestill remain: parallel transport, curvature, and holonomy.
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This example is from Wikipedia and may be reused under a CC BY-SA license.This yields a possible definition of an affine connection as acovariantderivativeor (linear) connection on the tangent bundle.
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This example is from Wikipedia and may be reused under a CC BY-SA license.In this language, an affine connection is simply acovariantderivativeor (linear) connection on the tangent bundle.
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This example is from Wikipedia and may be reused under a CC BY-SA license.By definition, acovariantderivativeof a scalar field is equal to the regular derivative of the field.
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This example is from Wikipedia and may be reused under a CC BY-SA license.These examples are from corpora and from sources on the web. Any opinions in the examples do not represent the opinion of the Cambridge Dictionary editors or of Cambridge University Press or its licensors.
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