next up previous contents index
Next: Pauli Matrices Up: Notation and Conventions Previous: Classification of 4-Vectors

Gradient and Differential Operators

The gradient vector operator is defined as $\vec{\nabla} \equiv (\partial/\partial x,\partial/\partial
y,\partial/\partial z)$.

The Laplacian scalar operator is defined as $\nabla^{2} \equiv
\vec{\nabla} \cdot \vec{\nabla}$.

The four partial-differential operators, $\partial/\partial x^\mu$, form a covariant vector, called the covariant gradient operator

...frac{1}{c} \frac{\partial}{\partial t},
\vec{\nabla}\right) .
\end{displaymath} (2.18)

Also, the contravariant gradient is defined as (notice the minus sign)

g^{\mu\nu}\partial_\nu\equiv\left( \frac{1}{c}
\frac{\partial}{\partial t}, -\vec{\nabla} \right).
\end{displaymath} (2.19)

Finally, the d'Alembertian is defined as

\Box \equiv \partial_\mu\partial^\mu \equiv \partial\cdot\pa...
...\equiv \frac{1}{c^2}\frac{\partial^2}{\partial
\end{displaymath} (2.20)

Sometimes the symbol $\triangle$ is used for the d'Alembertian.

Douglas M. Gingrich (gingrich@