The Jacobian ETSU（雅可比矩阵差）.pdf

The Jacobian The Jacobian of a Transformation 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; q i, 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. DeÖnition 5.1: A coordinate transformation T (u) is di§erentiable at a point p if there exists a matrix J (p) for which jjT (u) T (p) J (p) (u p)jj lim = 0 (1) u!p jju pjj When it exists, J (p) is the total derivative of T (u) at p. In non-vector notation, deÖnition 5.1 says that the total derivative at a point (p; q ) of a coordinate transformation T (u; v) is a matrix J (u; v) evaluated at (p; q ) : In a manner analogous to that in section 2-5, it can be shown that this matrix is given by J (u; v) = xu xv yu yv (see exercise 46). The total derivative is also known as the Jacobian Matrix of the transformation T (u; v) : EXAMPLE 1 What is the Jacobian matrix for the polar coordinate transformation? Solution: Since x = r cos () and y = r sin () ; the Jacobian matrix is xr x cos () r sin () J (r; ) = = yr y sin () r cos () If u (t) = hu (t) ; v (t)i is a curve in the uv-plan