 Solution

ID: past-exam-of-the-optics-and-electrodynamics-course-of-the-university-of-cambridge/2021/section-b/b6/a/iii/solution

\frac{\partial}{\partial x ^{' α}} = \frac{\partial x ^{β}}{\partial x ^{' α}} \frac{\partial}{\partial x ^{β}} .

Thus

\partial_{α} \equiv \frac{\partial}{\partial x ^{α}} transforms covariantly .

Differentiation with respect to a contravariant coordinate has a lower transformation index. In inertial Cartesian coordinates in Minkowski spacetime, no connection term is needed for this coordinate transformation of a derivative acting on a scalar; Lorentz transformations have constant Jacobians.

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