This derivation confirms a fundamental identity in rigid body dynamics, showing how the time derivative of the inertia tensor projected along angular velocity relates to a torque-like term involving the Levi-Civita symbol. Starting from the definition of the inertia tensor and using the rigid body velocity field $\vec{v}=\vec{\omega} \times \vec{x}$, we compute $I_{i j} \omega_j$ by differentiating under the integral and applying antisymmetric properties of the cross product. The derivation assumes rotation about the center of mass, which simplifies the expression by eliminating linear terms. Ultimately, both sides of the identity are shown to match, reinforcing the deep connection between rotational kinematics and the structure of the inertia tensor.

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✍️Mathematical Proof

$\complement\cdots$Counselor

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• Time Derivative of Inertia Tensor: The derivative $I_{i j}$ involves differentiating the position vectors inside the integral, using the product rule and the rigid body velocity expression.
• Contraction with Angular Velocity: Multiplying $I_{i j}$ by $\omega_j$ simplifies the expression and eliminates terms due to symmetry and center-of-mass conditions (e.g., $\int \rho x_i d V=0$ ).
• Use of Levi-Civita Symbol: The antisymmetric properties of $\varepsilon_{i j k}$ are crucial in simplifying terms and identifying vanishing contributions.
• Matching Both Sides: The right-hand side, $\varepsilon_{i j k} \omega_j I_{k \ell} \omega_{\ell}$, is expanded using the definition of $I_{k \ell}$, and shown to match the simplified left-hand side.
• Physical Interpretation: The identity expresses how the time rate of change of the inertia tensor, when projected along the angular velocity, relates to the torque-like term involving the cross product of angular momentum and angular velocity.
• Center-of-Mass Assumption: The derivation assumes the rotation is about the center of mass, ensuring that linear momentum terms vanish.

✍️Mathematical Proof

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1. Derivation of Tensor Transformation Properties for Mixed Tensors
2. The Polar Tensor Basis in Cartesian Form
3. Verifying the Rank Two Zero Tensor
4. Tensor Analysis of Electric Susceptibility in Anisotropic Media
5. Analysis of Ohm's Law in an Anisotropic Medium
6. Verifying Tensor Transformations
7. Proof of Coordinate Independence of Tensor Contraction
8. Proof of a Tensor's Invariance Property
9. Proving Symmetry of a Rank-2 Tensor
10. Tensor Symmetrization and Anti-Symmetrization Properties
11. Symmetric and Antisymmetric Tensor Contractions
12. The Uniqueness of the Zero Tensor under Specific Symmetry Constraints
13. Counting Independent Tensor Components Based on Symmetry
14. Transformation of the Inverse Metric Tensor
15. Finding the Covariant Components of a Magnetic Field
16. Covariant Nature of the Gradient
17. Christoffel Symbol Transformation Rule Derivation
18. Contraction of the Christoffel Symbols and the Metric Determinant
19. Divergence of an Antisymmetric Tensor in Terms of the Metric Determinant
20. Calculation of the Metric Tensor and Christoffel Symbols in Spherical Coordinates
21. Christoffel Symbols for Cylindrical Coordinates
22. Finding Arc Length and Curve Length in Spherical Coordinates
23. Solving for Metric Tensors and Christoffel Symbols
24. Metric Tensor and Line Element in Non-Orthogonal Coordinates
25. Tensor vs. Non-Tensor Transformation of Derivatives
26. Verification of Covariant Derivative Identities
27. Divergence in Spherical Coordinates Derivation and Verification
28. Laplace Operator Derivation and Verification in Cylindrical Coordinates
29. Divergence of Tangent Basis Vectors in Curvilinear Coordinates
30. Derivation of the Laplacian Operator in General Curvilinear Coordinates
31. Verification of Tensor Density Operations
32. Verification of the Product Rule for Jacobian Determinants and Tensor Density Transformation
33. Metric Determinant and Cross Product in Scaled Coordinates
34. Vanishing Divergence of the Levi-Civita Tensor
35. Curl and Vector Cross-Product Identity in General Coordinates
36. Curl of the Dual Basis in Cylindrical and Spherical Coordinates
37. Proof of Covariant Index Anti-Symmetrisation
38. Affine Transformations and the Orthogonality of Cartesian Rotations
39. Fluid Mechanics Integrals for Mass and Motion
40. Volume Elements in Non-Cartesian Coordinates (Jacobian Method)
41. Young's Modulus and Poisson's Ratio in Terms of Bulk and Shear Moduli
42. Tensor Analysis of the Magnetic Stress Tensor
43. Surface Force for Two Equal Charges
44. Total Electromagnetic Force in a Source-Free Static Volume
45. Proof of the Rotational Identity

🧄Proof and Derivation-1

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