🧄Metric Tensor and Line Element in Non-Orthogonal Coordinates

This problem beautifully illustrates how a non-orthogonal coordinate system impacts fundamental geometric measurements. The most important result is the non-zero off-diagonal term in the metric tensor, $g_{12}=1$, which is the defining characteristic of a non-orthogonal system, confirming that the new basis vectors are not perpendicular. Furthermore, the diagonal element $g_{22}=2$ shows the basis vector $E_2$ is not normalized (it has a length of $\sqrt{2}$ ). This non-trivial metric structure means that the formula for the length of a curve must include a cross-term $\left(2 \frac{d y^1}{d t} \frac{d y^2}{d t}\right)$, which accounts for the angle between the axes. If the system were Cartesian, this term would vanish, simplifying the line element back to the standard Pythagorean formula.

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