The metric tensor ( $g_{a b}$ ) is a matrix that defines distances in a coordinate system, with non-zero off-diagonal elements indicating non-perpendicular axes in a non-orthogonal system, like our skewed grid example, while its components can also vary with position, as seen with the $r^2$ term in polar coordinates, and its inverse $\left(g^{a b}\right)$ is crucial for raising/lowering indices but requires a more complex calculation than simple reciprocals, contrasting sharply with the identity matrix of a simple Cartesian system.
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$\complement\cdots$Counselor
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a non-orthogonal coordinate system dynamically calculating and displaying the metric tensor and its inverse
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$\gg$The Metric Tensor Covariant Derivatives and Tensor Densities
$\ggg$Mathematical Structures Underlying Physical Laws
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