What is Continuum Mechanics?
Continuum mechanics is a general term covering all those mechanics problems which can be described by functions that take values at all points in space. Hence fluid dynamics is a continuum mechanics problem, as we wish to find the velocity that describes the flow. This velocity takes the form of a function or field which has a defined value at any point in space (at least within that part of space that we are interested in). We could write this as
i.e. we are interested in the velocity field at any spatial coordinate (x,y,z) and at any time t. Other continuum mechanics problems include stress analysis in deformed bodies, and heat transfer problems. Continuum mechanics problems are usually described by complex sets of partial differential equations, for example the Navier-Stokes equations in fluid dynamics. These are typically only solvable using computational methods.
Some notes from a past course on the Theory of Elasticity are available.