Transfinite interpolations and Eulerian Lagrangian dynamics This book introduces transfinite interpolation as a generalization of interpolation of data prescribed at a finite number of points to data prescribed on a geometrically structured set such as a piece of curve surface or submanifold. The time invariant theory is readily extended to a moving deforming data set whose dynamics is specified in a Eulerian or Lagrangian framework. The resulting innovative tools cover a very broad spectrum of applications in fluid mechanics geometric optimization and imaging. The authors chose to focus on the dynamical mesh updating in fluid mechanics and the construction of velocity fields from the boundary expression of the shape derivative. Transfinite Interpolations and Eulerian Lagrangian Dynamics is a self contained graduate level text that integrates theory applications numerical approximations and computational techniques; and applies transfinite interpolation methods to finite element mesh adaptation and ALE fluid structure interaction. From finite to transfinite interpolations IDW and transfinite Barycentric interpolations Transfinite mean value interpolation Kriging and radial basis functions Dynamical transfinite interpolation Explicit moving mesh computations Moving mesh in monolithic fluid structure interaction Shape derivatives velocity method and interpolation Shape derivative via one sided derivatives of minima and minimax. In this book finite and transfinite interpolation methods are introduced at a general level in a unifying mathematical style