In differential geometry and topology one often deals with systems of partial differential equations, as well as partial differential inequalities, that have infinitely many solutions whatever boundary conditions are imposed. It was discovered in the fifties that the solvability of differential relations (i.e. equations and inequalities) of this kind can often be reduced to a problem of a purely homotopytheoretic nature. One says in this case that the corresponding differential relation satisfies the \(h\)principle. Two famous examples of the \(h\)principle, the NashKuiper \(C^1\)isometric embedding theory in Riemannian geometry and the SmaleHirsch immersion theory in differential topology, were later transformed by Gromov into powerful general methods for establishing the \(h\)principle. The authors cover two main methods for proving the \(h\)principle: holonomic approximation and convex integration. The reader will find that, with a few notable exceptions, most instances of the \(h\)principle can be treated by the methods considered here. A special emphasis in the book is made on applications to symplectic and contact geometry. Gromov's famous book "Partial Differential Relations", which is devoted to the same subject, is an encyclopedia of the \(h\)principle, written for experts, while the present book is the first broadly accessible exposition of the theory and its applications. The book would be an excellent text for a graduate course on geometric methods for solving partial differential equations and inequalities. Geometers, topologists and analysts will also find much value in this very readable exposition of an important and remarkable topic. Readership Graduate students and research mathematicians interested in global analysis and analysis on manifolds. Reviews "The reveiwed book is the first broadly accessible exposition of the theory written for mathematicians who are interested in an introduction into the \(h\)principle and its applications ... very readable, many motivations, examples and exercises are included ... a very good text for graduate courses on geometric methods for solving partial differential equations and inequalities."  Zentralblatt MATH "In my opinion, this is an excellent book which makes an important theory accessible to graduate students in differential geometry."  Jahresbericht der DMV Table of Contents Holonomic approximation  Jets and holonomy
 Thom transversality theorem
 Holonomic approximation
 Applications
Differential relations and Gromov's \(h\)principle  Differential relations
 Homotopy principle
 Open Diff \(V\)invariant differential relations
 Applications to closed manifolds
The homotopy principle in symplectic geometry  Symplectic and contact basics
 Symplectic and contact structures on open manifolds
 Symplectic and contact structures on closed manifolds
 Embeddings into symplectic and contact manifolds
 Microflexibility and holonomic \(\mathcal{R}\)approximation
 First applications of microflexibility
 Microflexible \(\mathfrak{U}\)invariant differential relations
 Further applications to symplectic geometry
Convex integration  Onedimensional convex integration
 Homotopy principle for ample differential relations
 Directed immersions and embeddings
 First order linear differential operators
 NashKuiper theorem
 Bibliography
 Index
