🧪Operations and Properties of Tensors

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.

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