📢Curl of the Dual Basis is Always Zero

The computation for the curl of the dual basis vectors in both cylindrical and spherical coordinates yields a null vector in every case. This fundamental result stems from the general tensorial expression for the curl, which is proportional to the partial derivative of the covariant components of the vector. Since the covariant components of the dual basis vector are given by the Kronecker delta, these components are constants. Consequently, their partial derivative is zero. This result is further verified when applying the physical component formula, where the term being differentiated, also simplifies to the constant, confirming that all components of the curl are zero in both coordinate systems.

Curl of the Dual Basis in Cylindrical and Spherical Coordinates | Proof and Derivationviadean on Notion