Lecture 5: Jacobians

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# Lecture 5: Jacobians - PowerPoint PPT Presentation

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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Presentation Transcript
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
• 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
• 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
• 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

Area of circle A=

r

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
• 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)