Vector calculus identities are confirmed through the interaction of symmetric partial derivatives and the antisymmetric Levi-Civita symbol. The first identity, $\nabla \times \nabla \phi=0$, relies on the commuting property of second-order derivatives, $\partial_j \partial_k \phi$, which creates a symmetric Hessian. Contracting this with the antisymmetric Levi-Civita symbol $\varepsilon_{i j k}$ results in zero. Similarly, the divergence of a curl, $\nabla \cdot(\nabla \times v)=0$, is proven by the symmetry of $\partial_i \partial_j v_k$. In both cases, relabeling dummy indices demonstrates that the expression equals its own negative, mathematically forcing the result to be zero.
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title Harmonic Identities: The Calculus of Physical Symmetry
Resulmation: Second-Order Vector Identities-Curl of Gradient and Divergence of Curl
IllustraDemo: Gravity Magnetism and Calculus Rules
Ex-Demo: The Harmonic Balance of Vector Fields
Narr-graphic: Vector Fields From Mathematical Symmetry to Physical Laws
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