🧄The Polar Tensor Basis in Cartesian Form

The analysis demonstrates that unlike the fixed Cartesian basis vectors, the polar basis vectors $E_r$ and $E_\theta$ are dynamic and change direction with the angle $\theta$. The core takeaway is the explicit transformation of the polar tensor basis ( $e_{a b}$ ) into the fixed Cartesian tensor basis ( $e_{i j}$ ). This is achieved by taking the outer product of the polar basis vectors, revealing that each of the four polar basis tensors ( $e_{r r}, e_{r \theta}, e_{\theta r}, e_{\theta \theta}$ ) is a linear combination of the Cartesian tensors. The coefficients of these combinations are directly dependent on trigonometric functions of $\theta$, which visually and mathematically confirms that the polar tensor basis rotates with its corresponding coordinate system.

The Polar Tensor Basis in Cartesian Form | Proof and Derivationviadean on Notion