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1、Lecture 5: Jacobians In 1D problems we are used to a simple change of variables, e.g. from x to u Example: Substitute 1D Jacobian maps strips of width dx to strips of width du 2D Jacobian For a continuous 1-to-1 transformation from (x,y) to (u,v) Then Where Region (in the xy plane) maps onto region
2、in the uv plane Hereafter call such terms etc 2D Jacobian maps areas dxdy to areas dudv Transformation T yield distorted grid of lines of constant u and constant v For small du and dv, rectangles map onto parallelograms This is a Jacobian, i.e. the determinant of the Jacobian Matrix Why the 2D Jacob
3、ian works The Jacobian matrix is the inverse matrix of i.e., Because (and similarly for dy) This makes sense because Jacobians measure the relative areas of dxdy and dudv, i.e So Relation between Jacobians Simple 2D Example r Area of circle A= Harder 2D Example where R is this region of the xy plane
4、, which maps to R here 1 4 8 9 An Important 2D Example Evaluate First consider Put as a-a -a a 3D Jacobian maps volumes (consisting of small cubes of volume .to small cubes of volume Where 3D Example Transformation of volume elements between Cartesian and spherical polar coordinate systems (see Lecture 4)
