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Abstract
This article analyzes the differential-geometric invariants of gradient vector fields. The properties of orthogonality to level surfaces, irrotationality, and harmonicity are presented with rigorous mathematical proofs, and their interrelationships are demonstrated. Three equivalent characterizations of a conservative field in simply connected domains are proven. The dimension-dependent structure of gradient fields is determined through the fundamental solutions of the Laplace equation. The applications of the results in electrostatics, hydrodynamics, and optimization theory are discussed.
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