# 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.

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