The Jacobian

Abstract

In this section, we explore the concept of a "derivative" of a coordinate transfor-mation, which is known as the Jacobian of the transformation. However, in this course, it is the determinant of the Jacobian that will be used most frequently. If we let u = hu; vi ; p = hp; qi, and x = hx; yi, then (x; y) = T (u; v) is given in vector notation by x = T (u) This notation allows us to extend the concept of a total derivative to the total derivative of a coordinate transformation. De{…}nition 5.1: A coordinate transformation T (u) is di¤ erentiable at a point p if there exists a matrix J (p) for which lim u!p jjT (u) T (p) J (p) (u p)jj jju pjj = 0 (1) When it exists, J (p) is the total derivative of T (u) at p.