Fractional Differential Geometry of Curves & Surfaces

Following the concepts of fractional differential and Leibnitz’s L-Fractional Derivatives, proposed by the author [1], the L-fractional chain rule is introduced. Furthermore, the theory of curves and surfaces is revisited, into the context of Fractional Calculus. The fractional tangents, normals, curvature vectors and radii of curvature of curves are defined. Moreover, the Serret-Frenet equations are revisited, into the context of fractional calculus. The proposed theory is implemented into a parabola and the curve configured by the Weierstrass function as well. The fractional bending problem of an inhomogeneous beam is also presented, as implementation of the proposed theory. Further, the theory is extended on manifolds, defining the fractional first differential (tangent) spaces, along with the revisiting first and second fundamental forms for the surfaces. In addition revisited operators like fractional gradient, divergence and rotation are introduced, outlining revision of the vector field theorems.