Oxford Calculus: Jacobians Explained

University of Oxford mathematician Dr Tom Crawford explains how to calculate the Jacobian for a 2D coordinate change and applies the general formula to polar coordinates.

Test yourself with some exercises on calculating Jacobians for parabolic, hyperbolic and spherical polar coordinates with this FREE worksheet in Maple Learn: https://learn.maplesoft.com/index.html#/?d=CFASOTKKPUFHMTBKCQPKFSCFKNCUIOFFESLHFUMMKNFIALEIMGBIOHLKPGCSMKEQHRIKBLLSFGAOAOEJDUEKJIHTCMNJFHFQAJBO

We begin with a discussion of when it is appropriate to change coordinates in an integral and how area calculations work in general. This is then exemplified with the unit circle and switching from Cartesian coordinates to polar coordinates where the Jacobian – or ‘stretch factor’ - is given by r.

We then derive the general formula for a 2D Jacobian using a geometrical approach and the deformation of a rectangle to a parallelogram. Finally, the general formula is used to verify the earlier result of the area of the unit circle being equal to pi.

Check your working using the Maple Calculator App – available for free on Google Play and the App Store.

Android: https://play.google.com/store/apps/details?id=com.maplesoft.companion&hl=en
Apple: https://apps.apple.com/us/app/maple-companion/id1466659419

Other videos in the Oxford Calculus series can be found here: https://www.youtube.com/playlist?list=PLMCRxGutHqflZoTY8JCm1GRzCdGXvZ3_S

Finding critical points for functions of several variables: https://youtu.be/Leomuu82-u8

Classifying critical points using the method of the discriminant: https://www.youtube.com/watch?v=5M_ts8Q2LEM

Partial differentiation explained: https://youtu.be/RVwcBGzQcT8

Second order linear differential equations: https://youtu.be/F54yhRB9qDI

Integrating factors explained: https://www.youtube.com/watch?v=ftqKuOfOX3E

Solving simple PDEs: https://youtu.be/uztjxrGY6Jw

Find out more about the Maple Calculator App and Maple Learn on the Maplesoft YouTube channel: https://www.youtube.com/channel/UCq2MmZQ8-kqEVAnmL2GSMsQ

Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: https://www.seh.ox.ac.uk/people/tom-crawford

For more maths content check out Tom's website https://tomrocksmaths.com/

You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths.

https://www.facebook.com/tomrocksmaths/
https://twitter.com/tomrocksmaths
https://www.instagram.com/tomrocksmaths/

Get your Tom Rocks Maths merchandise here:
https://beautifulequations.net/collections/tom-rocks-maths

Видео Oxford Calculus: Jacobians Explained канала Tom Rocks Maths