4-gradient

From the use of the Minkowski metric, converting between contravariant and covariant

$\mathbf x_{\mu} \equiv \eta_{\mu}^{\mu} \mathbf x^{\mu}$

Using matrix multiplication, we have already defined the covariant term,

$\mathbf{x_{\mu}}= \begin{bmatrix} dx_0 & -dx_1 & -dx_2 & -dx_3 \end{bmatrix}$

and the contravariant term

$\mathbf{x^{\mu}}= \begin{bmatrix} dx^0 \\ dx^1 \\ dx^2 \\ dx^3 \end{bmatrix}$

Following the rules of matrix multiplication this implies that the derivative with respect to a contravariant coordinate transforms as a covariant 4-vector.

$\nabla_{\mu}=\partial_{\mu}=\frac{\partial}{\partial x^{\mu}}$

$\mathbf \partial_\mu \equiv \Biggl [ \frac{\partial}{\partial t}\quad \frac{\partial}{\partial x}\quad \frac{\partial}{\partial y}\quad \frac{\partial}{\partial z}\Biggr ]=\Biggl [\frac{\partial}{\partial x^0}\quad \frac{\partial}{\partial x^1}\quad \frac{\partial}{\partial x^2}\quad \frac{\partial}{\partial x^3}\Biggr ]$