chapter  2
38 Pages

Calculus on manifolds

WithMichael K. Murray, John W. Rice

In Chapter 1 we introduced the idea of freeing a parametric family of probability distributions from its parameters, and treating the family as a geometric object like a surface. In this chapter we shall pursue the same idea in respect of differential calculus, and show how the theory of rates of change of functions can be freed from the notion of dependent and independent variables. However, co-ordinate systems or parametrisations, which are choices of independent variables, are not totally dispensable. They impose a certain minimal level of geometric structure on the set which they parametrise, in that they collectively determine the notion of tangent vector which is the foundation for describing rates of change of functions.