Tensor algebra, which includes vector-like operations such as addition and scalar multiplication, defines key products like the outer product, which creates a higher-rank tensor, and the contracted product, a generalization of the inner product formed by contraction, an operation that reduces rank by two. A tensor's inherent properties, like symmetry or anti-symmetry, allow for a unique decomposition of any rank-two tensor, while the Quotient Law is a crucial theorem used to verify if a mathematical object behaves as a true tensor.

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$\gg$Mathematical Structures Underlying Physical Laws

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• Tensor Algebra: Tensors can be added together and multiplied by a scalar, similar to vectors in a linear space.
• Outer Product: The outer product of two tensors, $T$ and $S$, creates a new tensor with a higher rank (the sum of their ranks).
• Contraction (Trace): This operation reduces the rank of a mixed tensor by two. It involves setting one contravariant and one covariant index equal and summing over them, which simplifies a tensor product into a lower-rank tensor.
• Contracted Product (Inner Product): A contracted product is a generalization of the inner product of vectors. It's formed by taking an outer product of two tensors and then performing one or more contractions.
• Symmetry and Anti-Symmetry: A tensor is symmetric if its components remain unchanged when a pair of indices is swapped ( $T^{a b}=T^{b a}$ ) and anti-symmetric if they change by a minus sign $\left(A^{a b}=-A^{b a}\right)$. These properties are inherent to the tensor itself and hold true in any coordinate system.
• Tensor Decomposition: Any rank-two tensor can be uniquely decomposed into a symmetric part and an anti-symmetric part. The number of independent components for these two parts adds up to the total number of components of a general tensor ( $N^2$ ).
• Quotient Law: This law states that if an object's components, when combined with a known tensor, produce another known tensor, then the object itself must also be a tensor. This is a crucial rule for verifying if a mathematical object is a tensor.

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1. The Outer Product and Tensor Transformations
2. Operations and Properties of Tensors
3. The Metric Tensor Covariant Derivatives and Tensor Densities
4. Tensors in Cartesian Coordinates and Their Integration
5. Applications of Tensors in Solid Mechanics Electromagnetism and Classical Mechanics

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1. calculate and display the angular velocity vector and the resulting angular momentum vector
2. the force being perpendicular to both the velocity and the field as result of the tensor's anti-symmetric nature
3. the stress tensor acts as a linear map that transforms the surface normal vector into the force vector
4. Visualize the outer product and contraction operations on tensors
5. how symmetric and anti-symmetric tensors behave by visualizing their effect on a sphere
6. The quotient law of tensors provides a test for whether a given set of components forms a tensor </aside>