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

$\complement\cdots$Counselor

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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.

✍️Mathematical Proof

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1. Proving the Cross Product Rules with the Levi-Civita Symbol
2. Proving the Epsilon-Delta Relation and the Bac-Cab Rule
3. Simplifying Levi-Civita and Kronecker Delta Identities
4. Dot Cross and Triple Products
5. Why a Cube's Diagonal Angle Never Changes
6. How the Cross Product Relates to the Sine of an Angle
7. Finding the Shortest Distance and Proving Orthogonality for Skew Lines
8. A Study of Helical Trajectories and Vector Dynamics
9. The Power of Cross Products: A Visual Guide to Precessing Vectors
10. Divergence and Curl Analysis of Vector Fields
11. Unpacking Vector Identities: How to Apply Divergence and Curl Rules
12. Commutativity and Anti-symmetry in Vector Calculus Identities
13. Double Curl Identity Proof using the epsilon-delta Relation
14. The Orthogonality of the Cross Product Proved by the Levi-Civita Symbol and Index Notation
15. Surface Parametrisation and the Verification of the Gradient-Normal Relationship
16. Proof and Implications of a Vector Operator Identity
17. Conditions for a Scalar Field Identity
18. Solution and Proof for a Vector Identity and Divergence Problem
19. Kinematics and Vector Calculus of a Rotating Rigid Body
20. Work Done by a Non-Conservative Force and Conservative Force
21. The Lorentz Force and the Principle of Zero Work Done by a Magnetic Field
22. Calculating the Area of a Half-Sphere Using Cylindrical Coordinates
23. Divergence Theorem Analysis of a Vector Field with Power-Law Components
24. Total Mass in a Cube vs. a Sphere
25. Momentum of a Divergence-Free Fluid in a Cubic Domain
26. Total Mass Flux Through Cylindrical Surfaces
27. Analysis of Forces and Torques on a Current Loop in a Uniform Magnetic Field
28. Computing the Integral of a Static Electromagnetic Field
29. Surface Integral to Volume Integral Conversion Using the Divergence Theorem
30. Circulation Integral vs. Surface Integral
31. Using Stokes' Theorem with a Constant Scalar Field
32. Verification of the Divergence Theorem for a Rotating Fluid Flow
33. Integral of a Curl-Free Vector Field
34. Boundary-Driven Cancellation in Vector Field Integrals
35. The Vanishing Curl Integral
36. Proving the Generalized Curl Theorem
37. Computing the Magnetic Field and its Curl from a Dipole Vector Potential
38. Proving Contravariant Vector Components Using the Dual Basis
39. Verification of Orthogonal Tangent Vector Bases in Cylindrical and Spherical Coordinates
40. Vector Field Analysis in Cylindrical Coordinates
41. Vector Field Singularities and Stokes' Theorem
42. Compute Parabolic coordinates-related properties
43. Analyze Flux and Laplacian of The Yukawa Potential
44. Verification of Vector Calculus Identities in Different Coordinate Systems
45. Analysis of a Divergence-Free Vector Field
46. The Uniqueness Theorem for Vector Fields
47. Analysis of Electric Dipole Force Field

🧄Proof and Derivation-2

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