A vector field, such as its expansion or compression (divergence) and its rotation (curl), are intrinsic to the field itself and don't depend on the coordinate system used to describe them. Even though the formulas for divergence and curl look vastly different in Cartesian, cylindrical, and spherical coordinates, applying them to the same vector field will always give the same result.

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✍️Mathematical Proof

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Consistency of Vector Identities

The core takeaway is that the fundamental properties of vector fields, such as their divergence (source/sink) and curl (rotation), are independent of the coordinate system used to describe them. Even though the mathematical expressions for divergence and curl are different in Cartesian, cylindrical, and spherical coordinates, applying them to the same vector field (in this case, the position vector $x$ ) consistently yields the same results: $\nabla \cdot x=3$ and $\nabla \times x=0$.

Geometric Interpretation

The analysis reinforces the geometric meaning of divergence and curl.

• Divergence: The position vector $x$ points radially outward from the origin. The fact that its divergence is a constant positive value (3) indicates that it acts as a uniform source everywhere. This makes sense-if you imagine a fluid flowing radially from the origin, its outward flow is consistent at every point.
• Curl: The curl of $x$ is zero everywhere, which signifies that the field has no rotational component. The position vector field is purely radial; it has no tendency to "circulate" around any point.

Power of Vector Calculus

The problem serves as a practical exercise in applying the coordinate-specific formulas for vector operators. Successfully performing the calculation in all three systems and confirming the consistency of the results demonstrates the robustness and elegance of vector calculus as a tool for describing physical phenomena.

✍️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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