In general relativity, a manifestly covariant equation is one in which all expressions are tensors.
Forbidden terms include but are not restricted to partial derivatives.
Tensor densities, especially integrands and variables of integration, may be allowed in manifestly covariant equations if they are clearly weighted by the appropriate power of the determinant of the metric.
If an equation is manifestly covariant, and if it reduces to a correct, corresponding equation in special relativity when evaluated instantaneously in a local inertial frame, then it is usually the correct generalization of the special relativistic equation in general relativity.
Note that the partial derivatives may be written in terms of covariant derivatives and Christoffel symbols as For a torsion-free metric assumed in general relativity, we may appeal to the symmetry of the Christoffel symbols which allows the field tensor to be written in manifestly covariant form