Rigid body motion is characterized by a total acceleration composed of tangential and centripetal components. A key property of rigid bodies is that the divergence of their velocity field is always zero, indicating incompressible motion where the body doesn't expand or contract. The curl of the velocity field is twice the angular velocity, illustrating the relationship between overall angular motion and the local swirling of particles within the body. Additionally, the divergence of the acceleration field is also always zero, directly stemming from its cross-product derivation.

<aside> 🧄

✍️Mathematical Proof

$\complement\cdots$Counselor

</aside>

The Total Acceleration has Two Components

A rigid body's total acceleration is a sum of two distinct parts: tangential acceleration, which arises from a change in the angular velocity ( $d \omega / d t$ ), and centripetal acceleration, which is always directed towards the axis of rotation and is responsible for the object's circular path.

Divergence of Velocity is Always Zero

The divergence ( $\nabla \cdot v$ ) of a rigid body's velocity field is always zero. This is a fundamental property of rigid body motion and signifies that the body is incompressible-it's not expanding or contracting at any point.

Curl of Velocity is Twice the Angular Velocity

The curl of the velocity field ( $\nabla \times v$ ) is always equal to twice the angular velocity ( $2 \omega$ ). This quantity measures the local rotation of the fluid or body and provides a direct link between the overall angular motion of the body and the microscopic swirling motion of its constituent particles.

Divergence of Acceleration is Always Zero

The divergence of the acceleration field ( $\nabla \cdot a$ ) is also always zero. This is a direct consequence of the fact that the acceleration is derived from cross products, similar to the velocity field.

✍️Mathematical Proof

‣

<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

</aside>