Some indentities for the gradient of the principal invariants, traces and determinants via Grassmann calculus.

Abstract

Let A be a second order tensor in a finite dimensional space. In this work we determine the gradient of the principal invariants ofAand obtain some trace and determinant identities using only some standard rigorous statements concerning Grassmann calculus. We recover some of the results of Dui et al. (J. Elast. 75:193–196, 2004) and of Truesdell and Noll (The Non-linear Field Theories of Mechanics, Springer, Berlin, 2002) and solve an old problem proposed in SIAM Review concerning a determinant identity from a new perspective in a concise and simple manner.