Since the first volume of this work came out in Germany in 1924, this book, together with its second volume, has remained standard in the field. Courant and Hilbert's treatment restores the historically deep connections between physical intuition and mathematical development, providing the reader with a unified approach to mathematical physics. The present volume represents Richard Courant's second and final revision of 1953.
From the Preface: (...) The book is addressed to students on various levels, to mathematicians, scientists, engineers. It does not pretend to make the subject easy by glossing over difficulties, but rather tries to help the genuinely interested reader by throwing light on the interconnections and purposes of the whole. Instead of obstructing the access to the wealth of facts by lengthy discussions of a fundamental nature we have sometimes postponed such discussions to appendices in the various chapters. Numerous examples and problems are given at the end of various chapters. Some are challenging, some are even difficult; most of them supplement the material in the text.
Courant and Friedrich's classical treatise was first published in 1948 and tThe basic research for it took place during World War II. However, many aspects make the book just as interesting as a text and a reference today. It treats the dynamics of compressible fluids in mathematical form, and attempts to present a systematic theory of nonlinear wave propagation, particularly in relation to gas dynamics. Written in the form of an advanced textbook, it should appeal to engineers, physicists and mathematicians alike.
Courant and Hilbert's treatment restores the historically deep connections between physical intuition and mathematical development, providing the reader with a unified approach to mathematical physics. · Transformation to Principal Axes of Quadratic and Hermitian Forms · Minimum-Maximum Property of Eigenvalues · Orthogonal Systems of Functions · Measure of Independence and Dimension Number · Fourier Series · Legendre Polynomials · The Expansion Theorem and Its Applications · Neumann Series and the Reciprocal Kernel · The Fredholm Formulas · Direct Solutions · The Euler Equations · Systems of a Finite Number of Degrees of Freedom · The Vibrating String · The Vibrating Membrane · Green's Function (Influence Function) and Reduction of Differential Equations to Integral Equations · Completeness and Expansion Theorems · Nodes of Eigenfunctions · Bessel Functions · Asymptotic Expansions