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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
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
2d jacobian
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

why the 2d jacobian works
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
relation between jacobians
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
simple 2d example
Simple 2D Example

Area of circle A=

r

harder 2d example

8

4

1

9

Harder 2D Example

where R is this region of the xy plane, which maps to R’ here

an important 2d example
An Important 2D Example
  • Evaluate
  • First consider
  • Put
  • as

a

a

-a

-a

3d jacobian
3D Jacobian
  • maps volumes (consisting of small cubes of volume
  • ........to small cubes of volume
  • Where
3d example
3D Example
  • Transformation of volume elements between Cartesian and spherical polar coordinate systems (see Lecture 4)