display the instantaneous line element at a moving point and compare the standard Cartesian length calculation to the one using the non-orthogonal metric

The metric tensor is the "correction factor" that translates skewed coordinate distances into true physical distances. The demonstration proves the central concept of general coordinate invariance: the calculated instantaneous length (speed) of the curve is identical whether computed using the simple, flat-space Cartesian formula ($\delta_{a b} \frac{d x^a}{d t} \frac{d x^b}{d t}$ ) or the complex non-orthogonal metric formula $(g_{a b} \frac{d y^a}{d t} \frac{d y^b}{d t})$. The Role of Off-Diagonal Terms: The fact that $g_{12}=1$ and $g_{22}=2$ in the non-orthogonal metric $(y^1, y^2)$ exactly compensates for the non-perpendicularity and non-unit length of the $y$-basis vectors. This compensation ensures that the final result remains the true Euclidean length, demonstrating that $g_{a b}$ precisely encodes the local geometry.

🎬Narrated Video

https://youtube.com/shorts/BosEnU97KbA?feature=share

🏗️Structural clarification of Poof and Derivation

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🗒️Downloadable Files - Recursive updates

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Copyright Notice

All content and images on this page are the property of Sayako Dean, unless otherwise stated. They are protected by United States and international copyright laws. Any unauthorized use, reproduction, or distribution is strictly prohibited. For permission requests, please contact [email protected]

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