Properties and application of smooth function germs of orbit tangent space

The finite determinacy of smooth function germ is the key in approximating the nonlinear function with infinite terms by its finite terms. In this paper, we discuss the inclusion relations with a new equivalent form for function germs in orbit tangent spaces, and get an improved form of the finite k -determinacy of smooth function germ. As an application, the methods in judging the right equivalency of Whitney function family with codimension 8 are presented.