four-gradient scalar field contravariant matrix
The four-gradient of a scalar field φ transforms covariantly rather than contravariantly:






(






1
c







∂
ϕ


∂

t
′











∂
ϕ


∂

x
′











∂
ϕ


∂

y
′











∂
ϕ


∂

z
′








)


=


(






1
c







∂
ϕ


∂
t










∂
ϕ


∂
x










∂
ϕ


∂
y










∂
ϕ


∂
z







)




(



γ


+
β
γ


0


0




+
β
γ


γ


0


0




0


0


1


0




0


0


0


1



)


.


{\displaystyle {\begin{pmatrix}{\dfrac {1}{c}}{\dfrac {\partial \phi }{\partial t'}}&{\dfrac {\partial \phi }{\partial x'}}&{\dfrac {\partial \phi }{\partial y'}}&{\dfrac {\partial \phi }{\partial z'}}\end{pmatrix}}={\begin{pmatrix}{\dfrac {1}{c}}{\dfrac {\partial \phi }{\partial t}}&{\dfrac {\partial \phi }{\partial x}}&{\dfrac {\partial \phi }{\partial y}}&{\dfrac {\partial \phi }{\partial z}}\end{pmatrix}}{\begin{pmatrix}\gamma &+\beta \gamma &0&0\\+\beta \gamma &\gamma &0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}.}


