A Better Calculus of Moving Surfaces

Abstract

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.