A divergence-free velocity field signifies an incompressible fluid where the density remains constant and mass is conserved, implying no fluid is being created or destroyed within a given volume. However, even in such a flow, total momentum is not necessarily zero. It can result from a non-zero average velocity, such as a constant upward velocity in the example provided. This leads to a net flow of momentum through the volume, demonstrating that divergence-free flows can still exhibit overall movement and momentum.

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✍️Mathematical Proof

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Divergence-Free Flow

• Zero Divergence, Constant Density: The analysis shows that the velocity field is divergence-free ( $\nabla \cdot v=0$ ). This is a crucial property for an incompressible fluid like the one described, where the density ( $\rho_0$ ) is constant.
• Conservation of Mass: A zero divergence mathematically confirms that there are no fluid sources or sinks within the volume. In physical terms, this means that fluid is neither being created nor destroyed, and the total mass within any given volume of the fluid remains constant over time. The fluid simply flows through the volume.

Total Momentum

• Non-Zero Momentum: Even though the flow is divergence-free, the total momentum of the fluid inside the cube is not zero. This is because the velocity field has a non-zero average velocity.
• Component Contributions: The swirling motion of the fluid in the $x^1-x^2$ plane contributes to the $P_1$ and $P_2$ components of the momentum. However, because the velocity is symmetric around the $z$-axis, the integral of these components across the cube results in equal and opposite values, making them non-zero in magnitude but equal and opposite in direction. The flow has a constant upward velocity ( $v_3=v_0$ ), which accumulates across the entire volume of the cube to create a significant total momentum in the $x^3$ direction.
• Net Flow: The resulting total momentum vector, $P$, indicates a net flow of momentum through the cube, primarily driven by the constant vertical velocity component.

✍️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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