This is accomplished by starting with the fundamental definition of the inverse metric tensor in terms of the dual basis vectors. By substituting the known transformation law for these vectors under a coordinate change, the derivation shows that the components in the new coordinate system that are related to the original components by the specific tensor transformation law. This result, with its two partial derivative terms in the numerator, is the hallmark of a contravariant tensor and proves the desired property.

<aside> <img src="/icons/profile_gray.svg" alt="/icons/profile_gray.svg" width="40px" />

$\complement\cdots$Counselor

</aside>

Transformation of the Inverse Metric Tensor

Transformation of the Inverse Metric Tensor

<aside> 📢

1. Symmetric and Antisymmetric Components
2. Symmetry and the Zero Tensor
3. Independent Components and Tensor Symmetry
4. Proof of the Contravariant Nature of the Inverse Metric Tensor
5. Christoffel Symbols Geometric Change in Cylindrical Coordinates
6. Geometric Meaning of Arc Length in Spherical Coordinates
7. Curvilinear Coordinate Systems and Their Metrics
8. Impact of Non-Orthogonal Coordinates on Geometric Measurement
9. Partial vs Covariant Derivative The Core Difference </aside>