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 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.

Key takeaways

Basis Vectors are Not Fixed : Unlike the Cartesian basis vectors $e_x$ and $e_y$, which are constant, the polar basis vectors $E_r$ and $E_\theta$ are a function of the angle $\theta$. They change direction at every point in space.
Outer Product Transformation : The tensor basis $e_{a b}=E_a \otimes E_b$ is formed by taking the outer product of these position-dependent basis vectors. This process transforms the simple vector basis into a more complex tensor basis.

The analysis explicitly breaks down how each polar basis tensor is composed of the Cartesian basis tensors $\left(e_{x x}, e_{x y}, e_{y x}, e_{y y}\right)$, revealing that each polar tensor is a linear combination of its Cartesian counterparts.

$e_{r r}$ : This tensor is defined by the components ( $\cos ^2 \theta, \sin \theta \cos \theta, \sin \theta \cos \theta, \sin ^2 \theta$ ) with respect to the Cartesian tensor basis. It is purely a combination of the radial component.
$e_{\theta 0}$ : This tensor is a combination of the same Cartesian basis tensors but with different coefficients, defined by ( $\sin ^2 \theta,-\sin \theta \cos \theta,-\sin \theta \cos \theta, \cos ^2 \theta$ ).
$e_{r \theta}$ and $e_{\theta r}$ : These mixed tensors represent the interaction between the radial and angular directions. Their components, such as ( $-\sin \theta \cos \theta, \cos ^2 \theta,-\sin ^2 \theta, \sin \theta \cos \theta$ ), highlight the relationship between the two different coordinate directions.

✍️Mathematical Proof

‣

<aside> 🧄

🧄Proof and Derivation-1

</aside>