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Lecture 5: Jacobians

Lecture 5: Jacobians. In 1D problems we are used to a simple change of variables, e.g. from x to u. 1D Jacobian maps strips of width d x to strips of width d u . Example: Substitute . 2D Jacobian.

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Lecture 5: Jacobians

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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 1D Jacobian maps strips of width dx to strips of width du • Example: Substitute

  2. 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 in the uv plane • Hereafter call such terms etc 2D Jacobian maps areas dxdy to areas dudv

  3. Why the 2D Jacobian works • 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

  4. Relation between Jacobians • The Jacobian matrix is the inversematrix of i.e., • Because (and similarly for dy) • This makes sense because Jacobians measure the relative areas of dxdy and dudv, i.e • So

  5. Simple 2D Example Area of circle A= r

  6. 8 4 1 9 Harder 2D Example where R is this region of the xy plane, which maps to R’ here

  7. An Important 2D Example • Evaluate • First consider • Put • as a a -a -a

  8. 3D Jacobian • maps volumes (consisting of small cubes of volume • ........to small cubes of volume • Where

  9. 3D Example • Transformation of volume elements between Cartesian and spherical polar coordinate systems (see Lecture 4)

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