🧄Curl of the Dual Basis in Cylindrical and Spherical Coordinates

The computation for the curl of the dual basis vectors ( $\nabla \times e^a$ ) in both cylindrical and spherical coordinates yields a null vector ( 0 ) 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, $\partial_b v_d$. Since the covariant components of the dual basis vector $e^a$ are given by the Kronecker delta, $v_d=\left(e^a\right)_d=\delta_d^a$, these components are constants (i.e., independent of the spatial coordinates). Consequently, their partial derivative is zero, meaning $\nabla \times e^a=0$. This result is further verified when applying the physical component formula, where the term being differentiated, $h_c \tilde{v}_c$, also simplifies to the constant $\delta_c^a$, 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