the covariant derivative is indispensable in non-Cartesian or curved systems that distinguishes it from the partial derivative

The animation vividly illustrates why the covariant derivative ( $\nabla V$ ) is indispensable in nonCartesian or curved systems, distinguishing it from the partial derivative ( $\partial V$ ). In the demo, the vector $V$ (blue arrow) is defined to have constant numerical components (e.g., $V^1=1.0$ ), meaning the partial derivative, which measures only component changes, is zero (orange arrow) and incorrectly reports no change. However, because the local basis vectors (black arrows) are intentionally rotating based on the point's position, the physical vector $V$ is forced to twist in space. The covariant derivative (the non-zero red arrow) accurately measures this true physical rate of twist, demonstrating that it is the only derivative capable of separating genuine physical change from the misleading artifacts caused by a changing coordinate system.

🎬Narrated Video

https://youtube.com/shorts/PwQsO_iudME?feature=share

🏗️Structural clarification of Poof and Derivation

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🗒️Downloadable Files - Recursive updates

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