= Ordinary covector derivative under nonlinear coordinate changes
Let $A'_b=J^d{}_b A_d$, where $J^d{}_b=\partial x^d/\partial x'^b$. Then
$$
\partial'_aA'_b=J^c{}_aJ^d{}_b\partial_cA_d+
\frac{\partial^2x^d}{\partial x'^a\partial x'^b}A_d.
$$
The second term prevents the ordinary derivative from being a generally <covariant tensor>. It vanishes for affine coordinate changes, including Cartesian <Lorentz transformations>, so the restricted claim of <Lorentz covariance> is valid. Antisymmetrization cancels the same term for arbitrary smooth coordinate changes and gives the <exterior derivative> of a <one-form>.
Back to article page
