Download presentation

Published byAmelia Trulove Modified over 3 years ago

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 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

5
Simple 2D Example Area of circle A= r

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

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)

Similar presentations

OK

For each point (x,y,z) in R3, the cylindrical coordinates (r,,z) are defined by the polar coordinates r and (for x and y) together with z. Example Find.

For each point (x,y,z) in R3, the cylindrical coordinates (r,,z) are defined by the polar coordinates r and (for x and y) together with z. Example Find.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Download ppt on world trade organization Ppt on edge detection in c Free download ppt on triangles for class 9 Ppt on conservation of angular momentum Ppt on review writing books Ppt on 3-phase squirrel cage induction motor Ppt on information technology and values Ppt on social etiquettes Ppt on finfet technology A ppt on water pollution