four-gradient

In differential geometry, the four-gradient (or 4-gradient) \boldsymbol{\partial} is the four-vector analogue of the gradient \vec{\boldsymbol{\nabla from vector calculus.

~38 min read

Article

22 sections

Contents

Notation
Definition
Usage
As a 4-divergence and source of conservation laws
As a Jacobian matrix for the SR Minkowski metric tensor
As a way to define the Lorentz transformations
As part of the total proper time derivative
As a way to define the Faraday electromagnetic tensor and derive the Maxwell equations
As a way to define the 4-wavevector
As the d'Alembertian operator
As a component of the 4D Gauss' Theorem / Stokes' Theorem / Divergence Theorem
As a component of the SR Hamilton–Jacobi equation in relativistic analytic mechanics
As a component of the Schrödinger relations in quantum mechanics
As a component of the covariant form of the quantum commutation relation
As a component of the wave equations and probability currents in relativistic quantum mechanics
As a key component in deriving quantum mechanics and relativistic quantum wave equations from special relativity
As a component of the RQM covariant derivative (internal particle spaces)
Derivation
See also
References
Note about References
Further reading

In differential geometry, the four-gradient (or 4-gradient) \boldsymbol{\partial} is the four-vector analogue of the gradient \vec{\boldsymbol{\nabla}} 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.