Final answer:
The covariant derivative of a rank 3 tensor remains a rank 3 tensor, keeping its rank unchanged, which is consistent with the principles of tensor calculus used in fields such as differential geometry and general relativity.
Step-by-step explanation:
The covariant derivative of a rank 3 tensor is still a rank 3 tensor. When you compute the covariant derivative of a tensor, you are effectively differentiating it with respect to the coordinate system while considering the manifold's connection, a concept from differential geometry often used in the field of general relativity and tensor calculus. Therefore, if you start with a rank 3 tensor and take its covariant derivative, you will still have a rank 3 tensor because the operation of taking the covariant derivative does not change the rank of the tensor. However, if it was indeed the question about how the components of the new tensor behave, then you would see an additional index appearing, associated with the derivative—effectively looking like a rank 4 tensor—but it's understood that the object itself remains a rank 3 tensor.