Seyyidoglu, M. S.; Tunçer, Y.; Uçar, D.; Berktaş, M. K.; Hatipoğlu, V. F. Forward curvatures on time scales. (English) Zbl 1220.53009 Abstr. Appl. Anal. 2011, Article ID 805948, 14 p. (2011). From the introduction: In recent years there have been a few research activities concerning the application of differential geometry on time scales. In [Turk. J. Math. 25, No. 4, 553–562 (2001; Zbl 0996.53002)], G. Sh. Guseinov and I. Özyılmaz defined the notions of forward tangent line, \(\Delta\)-regular curve, and natural \(\Delta\)-parametrization. Furthermore, in [J. Math. Anal. Appl. 326, No. 2, 1124–1141 (2007; Zbl 1118.26009], M. Bohner and G. Sh. Guseinov introduced the concept of a curve parametrized by a time scale parameter and gave integral formulas for computing its length in the plane. They established a version of the classical Green formula suitable to time scales. The general idea of this paper is to study forward curvature of curves where in the parametric equations the parameter varies in a time scale. We present the “differential” part of classical differential geometry on time scale calculus. The new results generalize the well-known formulas stated in classical differential geometry. We illustrate our results by applying them to various kinds of time scales. Cited in 3 Documents MSC: 53A04 Curves in Euclidean and related spaces 34A08 Fractional ordinary differential equations 26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems Keywords:forward tangent line; \(\Delta\)-regular curve; \(\Delta\)-parametrization; forward curvature Citations:Zbl 0996.53002; Zbl 1118.26009 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] G. Sh. Guseinov and E. Ozyılmaz, “Tangent lines of generalized regular curves parametrized by time scales,” Turkish Journal of Mathematics, vol. 25, no. 4, pp. 553-562, 2001. · Zbl 0996.53002 [2] M. Bohner and G. Sh. Guseinov, “Line integrals and Green’s formula on time scales,” Journal of Mathematical Analysis and Applications, vol. 326, no. 2, pp. 1124-1141, 2007. · Zbl 1118.26009 · doi:10.1016/j.jmaa.2006.03.040 [3] E. Ozyılmaz, “Directional derivative of vector field and regular curves on time scales,” Applied Mathematics and Mechanics (English Edition), vol. 27, no. 10, pp. 1349-1360, 2006. · Zbl 1167.53300 · doi:10.1007/s10483-006-1007-1 [4] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An introduction with Application, Birkhauser, Boston, Mass, USA, 2001. · Zbl 1107.34304 · doi:10.1016/S0898-1221(01)00189-4 [5] M. Bohner and G. Sh. Guseinov, “Partial differentiation on time scales,” Dynamic Systems and Applications, vol. 13, no. 3-4, pp. 351-379, 2004. · Zbl 1090.26004 [6] M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, Mass, USA, 2003. · Zbl 1025.34001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.