Basic physical laws and units -- Radiation balance of atmosphere -- Conservations in ocean and atmosphere -- Sobolev spaces I -- Particles and continuum mechanics -- Conservation of mass and momentum -- Conservation of energy -- One-dimensional wave equation -- Nonlinear effects, shocks -- Sobolev spaces II -- Linearized elasticity -- Ellipticity conditions -- Sobolev spaces III -- Sobolev spaces IV -- Sobolev spaces V -- Sobolev embedding theorem -- Fixed point theorems -- Brouwer's topological degree -- Time-dependent solutions I -- Time-dependent solutions II -- Time-dependent solutions III -- Uniqueness in 2 dimensions -- Traces -- Using compactness -- Existence of smooth solutions -- Semilinear models -- Size of singular sets -- Local estimates, compensated integrability -- Coriolis force -- Equation for the vorticity -- Boundary conditions in linearized elasticity -- Turbulence, homogenization -- G-convergence and H-convergence -- One-dimensional homogenization, Young measures -- Nonlocal effects I -- Nonlocal effects II -- A model problem -- Compensated compactness I -- Compensated compactness II -- Differential forms -- The compensated compactness method -- H-measures and variants -- Biographical Information -- Abbreviations and Mathematical Notation.