The solutions demonstrate how fundamental physical quantities in a continuous fluid are determined by integrating their respective densities over a given volume $V$. In all cases, the mass density $\rho(x)$ is crucial as it scales the quantity per unit mass to the quantity per unit volume, $d V$ . The total kinetic energy is a scalar, found by integrating the kinetic energy density $\frac{1}{2} \rho|v|^2$. In contrast, both the total momentum and total angular momentum are vector quantities. Total momentum is the integral of the linear momentum density $\rho v$. Total angular momentum, which must be defined relative to a specific reference point $x_0$, is the integral of the angular momentum density $\rho\left(x-x_0\right) \times v$.

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✍️Mathematical Proof

$\complement\cdots$Counselor

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• Total quantity as a Volume Integral: For continuous media like fluids, total physical quantities (like energy, momentum, or angular momentum) over a volume $V$ are found by integrating the corresponding density over that volume.
• Mass Density is the Multiplier: The mass density $\rho(x)$ serves as the fundamental measure of mass distribution and is the required factor in all integrals, multiplying the per-unit-mass quantity to get the per-unit-volume density.
• Kinetic Energy (Scalar): The integrand for kinetic energy is the scalar kinetic energy density $\frac{1}{2} \rho|v|^2$.
• Momentum and Angular Momentum (Vectors): Both total momentum and total angular momentum are vector quantities, meaning the integral itself yields a vector.
  • Momentum Density is the vector quantity $\rho v$.
  • Angular Momentum Density is the vector quantity $\rho\left(x-x_0\right) \times v$, which is the mass density times the cross product of the relative position vector and the velocity vector.
• Reference Point for Angular Momentum: Angular momentum is always defined relative to a specific reference point $x_0$. This is accounted for by using the relative position vector, $x-x_0$, in the integrand.

✍️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

🧄Proof and Derivation-1

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