The problem demonstrated how coordinate scaling affects the geometry of space, starting with the transformation $y^3=2 x^3$. This scaling leads to a diagonal metric tensor where only the $g_{33}$ component is altered, becoming 1 / 4, resulting in a metric determinant of $g=1 / 4$. The key implication is how this value scales the vector calculus operations: the Levi-Civita density $\eta^{a b c}$, crucial for the cross product, is scaled by $1 / \sqrt{g}=2$. Consequently, the contravariant components of the cross product, $(v \times w)^a=\eta^{a b c} v_b w_c$, are simply twice the magnitude of the standard Cartesian cross product involving the covariant components of the vectors, illustrating the general principle that all tensor operations in non-Cartesian coordinates must incorporate factors derived from the metric determinant.

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✍️Mathematical Proof

$\complement\cdots$Counselor

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1. Metric Tensor Calculation (Diagonal Metrics):

   • For an orthogonal coordinate transformation where $y^a$ only depends on $x^a$ (like scaling a single axis), the new metric tensor $g_{a b}$ is diagonal.
   • The diagonal components are determined by the scaling factors of the coordinate differentials:

   $$ g_{a a}=\sum_i\left(\frac{\partial x^i}{\partial y^a}\right)^2 $$

   • For the transformation $y^3=2 x^3$ (or $x^3=\frac{1}{2} y^3$ ), the metric component is $g_{33}= \left(\frac{1}{2}\right)^2=\frac{1}{4}$.

2. Metric Determinant and Volume Element:

   • The metric determinant $g$ quantifies how the local volume element is scaled. Since the metric was diagonal, the determinant is the product of the diagonal elements:

   $$ g=1 \cdot 1 \cdot \frac{1}{4}=\frac{1}{4} $$

   • The square root $\sqrt{g}=\frac{1}{2}$ relates the volume element in the new coordinates ( $\sqrt{g} d y^1 d y^2 d y^3$ ) to the Cartesian volume element ( $d x^1 d x^2 d x^3$ ).

3. Relating Levi-Civita Density and Symbol:

   • The Levi-Civita density $\left(\eta^{a b c}\right)$, used in tensor expressions for curl and cross products, is related to the standard Levi-Civita symbol $\left(\varepsilon^{a b c}\right)$ by the metric determinant:

   $$ \eta^{a b c}=\frac{1}{\sqrt{g}} \varepsilon^{a b c} $$

   • In this specific system, the density is $\eta^{a b c}=\frac{1}{1 / 2} \varepsilon^{a b c}=2 \varepsilon^{a b c}$. The scaling factor is 2 .

4. Cross Product in General Coordinates:

   • The contravariant components of the cross product $(v \times w)^a$ are calculated using the Levi-Civita density and the covariant components of the vectors:

   $$ (v \times w)^a=\eta^{a b c} v_b w_c $$

   • The result is a standard cross product formula multiplied by the scaling factor $1 / \sqrt{ g }= 2$:

   $$ (v \times w)^1= 2 \left( v _{ 2 } w _{ 3 }- v _{ 3 } w _{ 2 }\right) $$

✍️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

🧄Proof and Derivation-1

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