Transformation of the Inverse Metric Tensor

This is accomplished by starting with the fundamental definition of the inverse metric tensor in terms of the dual basis vectors, $g^{a b}=E^a \cdot E^b$. By substituting the known transformation law for these vectors under a coordinate change, the derivation shows that the components in the new coordinate system, $g^{l a b}$, are related to the original components by the specific tensor transformation law: $g^{\prime a b}=\frac{\partial x^a}{\partial x^{\prime c}} \frac{\partial x^b}{\partial x^{\prime a}} g^{c d}$. This result, with its two partial derivative terms in the numerator, is the hallmark of a contravariant tensor and proves the desired property.

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✍️Mathematical Proof

$\complement\cdots$Counselor

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The core of the proof is the definition of the inverse metric tensor components, $g^{a b}$, in terms of the dual basis vectors, $E^a$ and $E^b: g^{a b}=E^a \cdot E^b$. This equation links the tensor's components directly to the geometric properties of the dual basis.

The transformation of the dual basis vectors is the crucial step. They transform contravariantly, meaning their transformation rule involves the inverse Jacobian matrix of the coordinate transformation. This is represented by the formula $E^a=\frac{\partial x^a}{\partial x^b} E^{\prime b}$.

The derivation proceeds by substituting the dual basis vector transformation rule into the definition of the inverse metric tensor in the new coordinates, $g^{\prime a b}=E^{\prime a} \cdot E^{\prime b}$.

This equation is the definitive transformation rule for a contravariant tensor of rank 2. The two partial derivative terms, $\frac{\partial x^a}{\partial x^{\prime c}}$ and $\frac{\partial x^b}{\partial x^{\prime d}}$, are characteristic of a type (2,0) tensor and confirm its transformation properties.

✍️Mathematical Proof

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🧄Proof and Derivation-1

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