Transcription of Lecture 5: Jacobians - Rice University
1 Lecture 5: Jacobians In 1D problems we are used to a simple change of variables, from x to u Example: Substitute1D Jacobianmaps strips of width dxto strips of width du2D jacobian For a continuous 1-to-1 transformation from (x,y) to (u,v) Then Where Region (in the xy plane) maps onto region in the uv plane Hereafter call such terms etc2D Jacobianmaps areas dxdy toareas 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 , the determinant of the jacobian MatrixWhy the 2D jacobian works The jacobian matrix is the inverse matrix of , Because (and similarly for dy) This makes sense because Jacobians measure the relative areas ofdxdy and dudv, SoRelation between JacobiansSimple 2D ExamplerArea of circle A=Harder 2D Examplewhere R is thisregion of the xyplane, which maps toR here1489An Important 2D Example Evaluate First consider Put asa-a-aa3D jacobian maps volumes (consisting of small cubes of volume.)
2 To small cubes of volume Where3D Example Transformation of volume elements between Cartesian and spherical polarcoordinate systems (see Lecture 4)
