Mirela Ben Chen Technion - Israel Institute of Technology One of the main difficulties in shape analysis is shape matching, namely finding corresponding points between two shapes, often represented as surfaces embedded in 3D. This problem has been approached in many ways, yet a robust and globally applicable solution remains illusive. We will discuss a specific aspect of this wide problem, namely \emph{smooth self-maps}. In this case using \emph{tangent vector fields}, whose flows generate such maps, is instrumental, as the smoothness constraint is somewhat difficult to enforce. Thus, rephrasing mapping problems in terms of tangent vector fields has various advantages, even more so when employing the recently-proposed \emph{functional representation} of maps and vector fields. We will present this setup, and discuss various applications where analyzing maps through the lens of their vector field generators yields theoretical and practical advantages.