R.Witula et al obtained a stronger version of the second mean value theorem for integral with some restrictions.In this paper,the stronger version theorem is proved without any restriction.The result is first restrict...R.Witula et al obtained a stronger version of the second mean value theorem for integral with some restrictions.In this paper,the stronger version theorem is proved without any restriction.The result is first restricted to the Riemann integrable functions and can be easily generalized to L~p integrable functions by using the well-known result that continuous functions are dense in the Banach space L~p[a,b]for any p≥1.展开更多
By combining of the second gradient operator, the second class of integral theorems, the Gaussian-curvature-based integral theorems and the Gaussian （or spherical） mapping, a series of invariants or geometric conser...By combining of the second gradient operator, the second class of integral theorems, the Gaussian-curvature-based integral theorems and the Gaussian （or spherical） mapping, a series of invariants or geometric conservation quantities under Gaussian （or spherical） mapping are revealed. From these mapping invariants important transformations between original curved surface and the spherical surface are derived. The potential applications of these invariants and transformations to geometry are discussed展开更多
基金Supported by Natural Science Basic Research Program of Shaanxi(Program No.2021JM-487)the Special Scientific Research Program of the Education Department of Shaanxi Province(Grant No.18JK0161)the Scientific Research Foundation of Shaanxi University of Technology(Grant No.SLGQD1807)。
文摘R.Witula et al obtained a stronger version of the second mean value theorem for integral with some restrictions.In this paper,the stronger version theorem is proved without any restriction.The result is first restricted to the Riemann integrable functions and can be easily generalized to L~p integrable functions by using the well-known result that continuous functions are dense in the Banach space L~p[a,b]for any p≥1.
基金Project supported by the National Natural Science Foundation of China (No.10572076)
文摘By combining of the second gradient operator, the second class of integral theorems, the Gaussian-curvature-based integral theorems and the Gaussian （or spherical） mapping, a series of invariants or geometric conservation quantities under Gaussian （or spherical） mapping are revealed. From these mapping invariants important transformations between original curved surface and the spherical surface are derived. The potential applications of these invariants and transformations to geometry are discussed
