We give a proof of Green’s theorem which captures the underlying intuition and which relies only on the mean value theorems for derivatives and integrals and on the change of variables theorem for double integrals.

We study diamond-alpha integrals on time scales. A diamond-alpha version of Fermat’s theorem for stationary points is also proved, as well as Rolle’s, Lagrange’s, and Cauchy’s mean value theorems on time scales. Mathematics Subject Classification 2000: 26A42; 39A12.

We establish improved mean value estimates associated with the number of integer solutions certain systems diagonal equations, in some instances attaining sharpest conjectured conclusions. This is first occasion on which bounds this quali