A better calculus of moving surfaces. (English) Zbl 1428.53013

Summary: We introduce \(\dot{\nabla}\), a new invariant time derivative with respect to a moving surface that is a modification of the classical \(\delta/\delta\)-derivative. The new operator offers significant advantages over its predecessor. In particular, it produces zero when applied to the surface metric tensors \(S_{\alpha \beta}\) and \(S^{\alpha \beta}\) and therefore permits free juggling of surface indices in the calculus of moving surfaces identities. As a result, the table of essential differential relationships is cut in half. To illustrate the utility of the operator, we present a calculus of moving surfaces proof of the Gauss-Bonnet theorem for smooth closed two dimensional hypersurfaces.


53A05 Surfaces in Euclidean and related spaces
76A20 Thin fluid films