🎬display the instantaneous line element at a moving point and compare the standard Cartesian length c

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.

Metric Tensor and Line Element in Non-Orthogonal Coordinates | Proof and Derivationviadean on Notion