🧄Commutativity and Anti-symmetry in Vector Calculus Identities

The fundamental vector calculus identities-that the curl of a gradient and the divergence of a curl are zero-are confirmed by the perfect cancellation between the commutative properties of partial derivatives and the antisymmetry of the Levi-Civita symbol, while in a separate context, a point charge's electric field and potential, which decreases as 1 / r, satisfies Laplace's equation away from the charge, illustrating the foundational relationship between field, potential, and charge distribution in electrostatics.

🎬Visualize the radial electric field and the potential and the Laplacian of the potential

The electric field of a point charge is radially outward and inversely proportional to the square of the distance, while its potential decreases as 1/r. Away from the charge, the potential satisfies Laplace’s equation, illustrating the fundamental link between field, potential, and charge distribution in electrostatics.

🖊️Mathematical Proof

Commutativity and Anti-symmetry in Vector Calculus Identities | Proof and Derivationviadean on Notion