Four-gradient

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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 $\Box$

[edit] 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