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Four-gradient

 
Wikipedia: Four-gradient

The four-gradient is the four-vector generalization of the gradient:

\partial_\alpha \ \stackrel{\mathrm{def}}{=}\  \left(\frac{1}{c} \frac{\partial}{\partial t}, \nabla \right)

and is sometimes also represented as D.

The square of D is the four-Laplacian, which is called the d'Alembert operator:

D\cdot D = \partial_\alpha \partial^\alpha = \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \nabla^2 .

As it is the dot product of two four-vectors, the d'Alembertian is a Lorentz invariant scalar.

It is also written \Box

References

  • S. Hildebrandt, "Analysis II" (Calculus II), ISBN 3-540-43970-6, 2003
  • L.C. Evans, "Partial differential equations", A.M.Society, Grad.Studies Vol.19, 1988
  • J.D. Jackson, "Classical Electrodynamics" Chapter 11, Wiley ISBN 0-471-30932-X

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Wikipedia. This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article "Four-gradient" Read more