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.

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✍️Mathematical Proof

$\gg$Mathematical Structures Underlying Physical Laws

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• Commutativity and Antisymmetry: The core of both proofs relies on the interaction between two fundamental properties: the commutativity of partial derivatives and the antisymmetry of the Levi-Civita symbol.
• Proof for $\nabla \times \nabla \phi=0$ : This identity, stating that the curl of a gradient is always zero, holds true because the symmetry of the second partial derivatives $\left(\partial_j \partial_k \phi=\partial_k \partial_j \phi\right)$ perfectly cancels out the antisymmetry of the Levi-Civita symbol ( $\varepsilon_{i j k}=-\varepsilon_{i k j}$ ) in the summation. For every term with positive Levi-Civita coefficient, there is an equal and opposite term with a negative coefficient, causing the entire sum to vanish.
• Proof for $\nabla \cdot(\nabla \times v)=0$ : Similarly, the divergence of the curl of a vector field is zero. This proof works for the same reason: the symmetry of the second partial derivatives of the vector's components $\left(\partial_i \partial_j v^k=\partial_j \partial_i v^k\right)$ directly opposes the antisymmetry of the LeviCivita symbol, leading to complete cancellation of terms in the summation.

In essence, these two identities confirm that the vector calculus operations of curl and divergence are a perfect match for the properties of the Levi-Civita symbol, ensuring that these complex combinations always simplify to zero.

🎬Demonstration

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 ($\nabla^2 \phi = 0$), illustrating the fundamental link between field, potential, and charge distribution in electrostatics.

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

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

✍️Mathematical Proof

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Proving the Cross Product Rules with the Levi-Civita Symbol

Proving the Epsilon-Delta Relation and the Bac-Cab Rule

Simplifying Levi-Civita and Kronecker Delta Identities

Dot Cross and Triple Products

Why a Cube's Diagonal Angle Never Changes

How the Cross Product Relates to the Sine of an Angle

Finding the Shortest Distance and Proving Orthogonality for Skew Lines

A Study of Helical Trajectories and Vector Dynamics

The Power of Cross Products: A Visual Guide to Precessing Vectors

Divergence and Curl Analysis of Vector Fields

Unpacking Vector Identities: How to Apply Divergence and Curl Rules

Commutativity and Anti-symmetry in Vector Calculus Identities

Double Curl Identity Proof using the epsilon-delta Relation

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