A vector field with zero curl can still have a non-zero circulation integral if the integration path encloses a singularity, which is a point where the field is undefined. This demonstrates a crucial exception to Stokes' Theorem, which assumes the absence of such singularities within the surface of integration.

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✍️Mathematical Proof

$\gg$Mathematical Structures Underlying Physical Laws

$\complement\cdots$Counselor

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Divergence and Curl

The vector field $v=\frac{1}{\rho} e_\phi$ has both a divergence of zero and a curl of zero for all $\rho>0$. This means the field is "source-free" and "irrotational" in this region.

Non-zero Circulation Integral

Despite having a curl of zero, the line integral of the vector field around a closed loop is non-zero ( $4 \pi$ ). This seemingly contradictory result is because the vector field has a singularity at $\rho=0$, which is enclosed by the path of integration.

Stokes' Theorem

The analysis demonstrates a key exception to Stokes' Theorem. The theorem, which states that the circulation of a vector field is equal to the flux of its curl through a surface, only applies to surfaces that do not contain singularities of the vector field. The non-zero result of the integral, despite the zero curl, highlights the presence of a singularity at the origin.

Simplified Integration

The direct calculation of the line integral was simplified by using the property of orthogonal basis vectors, which allowed the dot product $v \cdot d x$ to be easily reduced to just $d \phi$. This streamlined the process of solving the integral along the given path.

✍️Mathematical Proof

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<aside> 🧄

  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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