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.

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✍️Mathematical Proof

Counselor

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• 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

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1. Derivation of Tensor Transformation Properties for Mixed Tensors
2. The Polar Tensor Basis in Cartesian Form
3. Verifying the Rank Two Zero Tensor
4. Tensor Analysis of Electric Susceptibility in Anisotropic Media
5. Analysis of Ohm's Law in an Anisotropic Medium

🧄Proof and Derivation-1

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