# Ditch the Messy Math The Elegant Tensor Secret to Centrifugal Force

The fictitious centrifugal force, initially defined by the vector triple product $$\vec{F}{c}=m \vec{\omega} \times(\vec{\omega} \times \vec{x})$$, can be elegantly transformed into a linear tensor expression, $$F{c}^i=T^{i j} x^j$$. By applying the standard vector triple product identity, the force is expanded to $$\vec{F}\_c=m\left\[\vec{\omega}(\vec{\omega} \cdot \vec{x})-\omega^2 \vec{x}\right]$$, which allows for the direct identification of the components of the tensor as $$T^{i j}=m\left(\omega^i \omega^j-\omega^2 \delta^{i j}\right)$$, where $$\omega^2=\omega^k \omega^k$$ and $$\delta^{i j}$$ is the Kronecker delta. This tensor $$T^{i j}$$ is symmetric and depends quadratically on the angular velocity $$\vec{\omega}$$, effectively encoding the force's linear dependence on the position vector $$\vec{x}$$. Physically, the centrifugal force vanishes if and only if the displacement vector $$\vec{x}$$ is parallel to the angular velocity $$\vec{\omega}$$, meaning the particle lies along the axis of rotation.

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