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

$$\partial_\mu\equiv\frac{\partial}{\partial x^\mu}\equiv\left... ...frac{1}{c} \frac{\partial}{\partial t}, \vec{\nabla}\right) .$$ (2.18)

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

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

Finally, the d'Alembertian is defined as

$$\Box \equiv \partial_\mu\partial^\mu \equiv \partial\cdot\pa... ...\equiv \frac{1}{c^2}\frac{\partial^2}{\partial t^2}-\nabla^2.$$ (2.20)

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