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4-gradient
From the use of the Minkowski metric, converting between contravariant and covariant
[math]\mathbf x_{\mu} \equiv \eta_{\mu}^{\mu} \mathbf x^{\mu}[/math]
Where we have already defined the covariant term,
[math]\mathbf{x_{\mu}}= \begin{bmatrix}
x_0 & -x_1 & -x_2 & -x_3
\end{bmatrix}[/math]
and the contravariant term
[math]\mathbf{x^{\mu}}=
\begin{bmatrix}
x^0 \\
x^1 \\
x^2 \\
x^3
\end{bmatrix}
[/math]
From quantum mechanics we know that partial differential is a linear operator. Following the rules of matrix multiplication this implies that the derivative with respect to a contravariant coordinate transforms as a covariant 4-vector, and the derivative with respect to a covariant coordinate transforms as a contravariant vector.
[math]\partial_{\mu}=\frac{\partial}{\partial x^{\mu}}[/math]
[math]\mathbf \partial_\mu \equiv \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 ]=\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 t}\quad -\nabla \Biggr ][/math]
[math]\partial^{\mu}=\frac{\partial}{\partial x_{\mu}}[/math]
[math]\mathbf \partial^\mu \equiv
\begin{bmatrix}
\frac{\partial}{\partial x_0} \\
\\
\frac{\partial}{\partial x_1} \\
\\
\frac{\partial}{\partial x_2} \\
\\
\frac{\partial}{\partial x_3}
\end{bmatrix}=
\begin{bmatrix}
\frac{\partial}{\partial t} \\
\\
\frac{\partial}{\partial x} \\
\\
\frac{\partial}{\partial y} \\
\\
\frac{\partial}{\partial z}
\end{bmatrix}=
\begin{bmatrix}
\frac{\partial}{\partial t} \\
\nabla
\end{bmatrix}
[/math]
Since it is an operator, the dot product of two partial differentials yields an operator known as the D'Alembert operator.
[math]\partial^{\mu} \partial_{\mu}=
\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 ]\cdot
\begin{bmatrix}
\frac{\partial}{\partial x_0} \\
\\
\frac{\partial}{\partial x_1} \\
\\
\frac{\partial}{\partial x_2} \\
\\
\frac{\partial}{\partial x_3}
\end{bmatrix}=
\frac{\partial^2}{\partial t^2}-\nabla^2\equiv \Box
[/math]
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