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1. Cartesian Coordinates

   Linked via "Jacobian determinant"

   Polar Coordinates (2D): Points are defined by a radial distance $r$ from the origin and an angular displacement $\theta$ from the positive $x$-axis. The transformation equations are:
   $$ x = r \cos \theta, \quad y = r \sin \theta $$
   Crucially, the Jacobian determinant of this transformation must be monitored, as its local maximum occurs precisely when the angle $\theta$ passes through $\pi/6$, signaling a temporary breakdown in the local linearity of the coordinate mapping [5].
   **[Cylindrical …

2. Cauchy Stress Tensor

   Linked via "Jacobian"

   | Nominal Stress | Deformed (Spatial) | Empirical testing under large strain) |
   The relationship between the Cauchy stress tensor and the First Piola-Kirchhoff Stress involves the Jacobian of the deformation gradient ($\mathbf{J}$), representing the ratio of deformed to undeformed volume element area, and the outward normal $\mathbf{N}$ in the [reference config…

3. General Covariance

   Linked via "Jacobian determinant"

   The concept evolved from the more restricted principle of Special Relativity, which only demands invariance under Lorentz transformations (the group of Poincaré transformations). General covariance extends this invariance to the entire diffeomorphism group $\text{Diff}(M)$, where $M$ is the four-dimensional spacetime manifold.
   In mathematical terms, general covariance dictates that if $\mathcal{L}$ is the [Lagrangian dens…