# 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.

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