ABSTRACT
Manifolds provide a generalization to the concept of a curve or a surface, objects introduced in the usual calculus sequence. In calculus of a real variable, one does not study functions defined over a discrete set of real values because the notions behind continuity and differentiability do not make sense over such sets. Intuitively, a function is called continuous if it preserves “nearness.” One important aspect of critical points already arises with real functions. In multivariable calculus or in a basic differential geometry course, one typically uses yet another technique to visualize functions of the form multivariable functions. Out of the vast variety of possible functions one could study, the class of linear functions serves a fundamental role in the analysis of multivariable functions. A particularly important case of matrices representing linear transformations is the change of basis matrix.
