A Stokes Theorem for Second Order Tensor Fields and its Implications in Continuum Mechanics * Roger Fosdick Department of Aerospace Engineering and Mechanics University of Minnesota, Minneapolis, MN 55455, USA Gianni Royer-Carfagni Department of Civil-Environmental Engineering and Architecture University of Parma, Parco Area delle Scienze 181/A, I 43100 Parma, Italy July 15, 2004 Abstract We give a constructive proof of a particular Stokes theorem (1.4) for tensor fields in R 3 ⊗ R 3 . Its specialization to symmetric tensor fields, given in (1.5), bears a close relation to compatibility in linear elasticity theory and to the generalized Beltrami representation of symmetric tensor fields in continuum mechanics. These issues are discussed. Keywords: Stokes theorem, compatibility, Ces`aro representation, Beltrami represen- tation. Dedicated to Niall Horgan on the occasion of his sixtieth birthday. 1 Introduction: A particular Stokes theorem. Given a bounded, open and connected set D∈ R 3 , a smooth tensor field S(x)= S ij (x)i i ⊗ i j ∈ R 3 ⊗ R 3 for x ∈D and an arbitrary point z ∈D, let us define K(x, z)= K ij (x, z)i i ⊗ i j ∗ We gratefully acknowledge the partial support of the Minnesota Supercomputing Institute, the Italian M.U.R.S.T. through the project ‘Mathematical Models for Materials Science’ within the research activities of the EU Network ‘Phase Transitions in Crystalline Solids’ and the National Science Foundation Grant No. DMS-0102841. 1