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.
General Result for Dual Basis Curl: The curl of a dual basis vector $\left(\nabla \times e^a\right)$ in any orthogonal curvilinear coordinate system is always zero.
Reasoning from General Curl Expression: Using the general expression for the contravariant components of the curl, $(\nabla \times v)^c=\frac{1}{\sqrt{g}} \varepsilon^{c a b} \partial_a v_b$, and noting that the covariant components of $e^a$ are $v_b=\left(e^a\right)_b=\delta_b^a$, the partial derivative term $\partial_a \delta_b^a$ is zero because $\delta_b^a$ is constant.
Result in Specific Coordinate Systems: Consequently, the physical components of the curl of the dual basis are all zero in both:
Verification with known formula: The result is confirmed by using the physical component expression. The physical components of $e^a$ are $\tilde{v}_c=\left(e^a\right)_c=\frac{1}{h_c} \delta_c^a$. The term being differentiated, $h_c \tilde{v}_c$, simplifies to $\delta_c^a$, whose derivative is zero.
$$ \partial_b\left(h_c \tilde{v}_c\right)=\partial_b\left(\delta_c^a\right)=0 $$
This mathematically validates the zero result in the physical component framework as well.
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