Transformation Laws: Diagonalization of Symmetric Tensors

This lecture covers the transformation of tensor components and the diagonalization of symmetric tensors, focusing on the principles of objectivity and the definition of tensors. The instructor explains how to map tensor components from one basis to another, emphasizing the importance of transformation laws for tensors of various orders. The lecture includes a detailed discussion on the addition and multiplication rules for tensors, demonstrating how these rules confirm the definition of tensors through transformation behavior. The analysis progresses to the symmetric stress tensor, highlighting the significance of principal stresses and the characteristic equation. The instructor illustrates how to find a diagonal basis for symmetric tensors, ensuring the absence of shear stresses. This foundational knowledge is crucial for understanding material behavior under stress and is linked to concepts from linear algebra, such as eigenvalues and eigenvectors. The lecture concludes with a review of the invariants of the stress tensor, which are essential for characterizing its properties across different coordinate frames.