The analysis of the divergence-free vector field $v$ and its vector potential $A$ visually and mathematically demonstrates a fundamental principle of vector calculus: a field with no sources or sinks can be fully described by a second field-its vector potential-whose curl generates the first field. This shows that the rotational nature of $A$ gives rise to the outward flow of $v$, linking the two seemingly distinct properties in an elegant and consistent way.

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✍️Mathematical Proof

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Verifying the Vector Field is Divergence-Free

The core takeaway is that the vector field $v=\frac{1}{r^2} e_r$ is divergence-free $(\nabla \cdot v=0)$ for all points where $r>0$. This property is fundamental to the field's behavior. A divergence-free field is one that has no sources or sinks; in other words, what flows into a given volume must flow out, leaving no net change.

The mathematical proof relies on applying the divergence operator in spherical coordinates, which is the most natural coordinate system for this vector field due to its radial symmetry.

Finding the Vector Potential

The analysis successfully demonstrates how to find a vector potential $A$ for a divergence-free field. The existence of a vector potential is guaranteed by the property that $\nabla \cdot v=0$. The key steps are:

• Solve the Curl Equation: By setting $v=\nabla \times A$, we used the given form of $A$ to establish a relationship between the components of $v$ and the unknown $A_{\varphi}$. This led to an expression for $A_{\varphi}$ in terms of $r$ and $\theta$.
• Verify Divergence of the Potential: The final check confirmed that the derived vector potential, $A=-\frac{\cot \theta}{r} e_{\varphi}$, is also divergence-free ( $\nabla \cdot A=0$ ). This is an important consistency check, as it ensures the potential itself does not have any sources or sinks, a common requirement in many physical applications.

✍️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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