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This is the fourth of a five-volume exposition of the main principles of nonlinear functional analysis and its applications to the natural sciences, economics, and numerical analysis. The presentation is self-contained and accessible to the nonspecialist. Topics covered in this volume include applications to mechanics, elasticity, plasticity, hydrodynamics, thermodynamics, stastical physics, and special and general relativity including cosmology. The book contains a detailed physical motivation of the relevant basic equations and a discussion of particular problems which have played a significant role in the development of physics and through which important mathematical and physical insight may be gained. An attempt is made to combine classical and modern ideas and to build a bridge between the language and thoughts of physicists and mathematicians. Many exercises and a comprehensive bibliography complement the text. This corrected printing contains many revisions as well as a list of new and recent references.