Partial differential equations Michael E. Taylor ; 1: Basic theory The first of three volumes on partial differential equations this one introduces basic examples arising in continuum mechanics electromagnetism complex analysis and other areas and develops a number of tools for their solution in particular Fourier analysis distribution theory and Sobolev spaces. These tools are then applied to the treatment of basic problems in linear PDE including the Laplace equation heat equation and wave equation as well as more general elliptic parabolic and hyperbolic equations. The book is targeted at graduate students in mathematics and at professional mathematicians with an interest in partial differential equations mathematical physics differential geometry harmonic analysis and complex analysis. The third edition further expands the material by incorporating new theorems and applications throughout the book and by deepening connections and relating concepts across chapters. In includes new sections on rigid body motion on probabilistic results related to random walks on aspects of operator theory related to quantum mechanics on overdetermined systems and on the Euler equation for incompressible fluids. The appendices have also been updated with additional results ranging from weak convergence of measures to the curvature of Kahler manifolds.