Tensors are mathematical objects that are defined by their transformation properties under a change of coordinates. This means that if a quantity transforms according to a specific set of rules involving the partial derivatives of the coordinate systems, it's considered a tensor. Based on this definition, we can verify that several expressions are indeed tensors. For example, multiplying a tensor by a scalar, adding two tensors of the same rank, and taking the outer product of two tensors all result in a new quantity that also transforms according to the correct tensor rules, thus demonstrating that these operations preserve the tensor nature of the quantities involved.

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✍️Mathematical Proof

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• Tensor Definition: A tensor is defined by how its components transform under a change of coordinates. The specific transformation rule for a tensor of type ( $n, m$ ) involves a product of partial derivatives: $n$ for the contravariant indices and $m$ for the covariant indices.
• Scalar Multiplication: Multiplying a tensor by a scalar (a rank-0 tensor) results in a new tensor of the same rank. The transformation factors are the same for both the original tensor and the new one, and they simply multiply the scalar value.
• Tensor Addition: The sum of two tensors of the same rank is also a tensor of that rank. This is because the distributive property of multiplication allows the common transformation factors to be factored out, leaving the sum of the original tensor components.
• Outer Product: The outer product of two tensors is a new tensor whose rank is the sum of the ranks of the original tensors. For example, the outer product of a rank-( 1,1 ) tensor and a rank- $(0,2)$ tensor results in a rank- (1,3) tensor. The transformation rule for the new tensor is simply the product of the individual transformation rules of the original tensors.

✍️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
6. Verifying Tensor Transformations
7. Proof of Coordinate Independence of Tensor Contraction
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🧄Proof and Derivation-1

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