❓What is the form of the centrifugal force in component notation before identifying the tensor?

The centrifugal force in component notation is: $F_c^i=m\left[\omega^i\left(\omega^j x^j\right)-\omega^2 x^i\right]$. This formula takes the difference between the i-th component of two terms-one parallel to $\omega$ (Term 1) and one parallel to x (Term 2)-multiplies the result by the mass m, and yields the i-th component of the centrifugal force. This rearrangement isolates the dependence on the position components ( $x^j$ and $x^i$ ) so the coefficient of x can be cleanly identified as the centrifugal tensor $T^{i j}$.

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