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/doc/yt882...

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.

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Other videos in the Oxford Calculus series can be found here:    • Oxford Calculus  

Finding critical points for functions of several variables:    • Oxford Calculus: Finding Critical Poi...  

Classifying critical points using the method of the discriminant:    • Oxford Calculus: Classifying 2D Criti...  

Partial differentiation explained:    • Oxford Calculus: Partial Differentiat...  

Second order linear differential equations:    • Oxford Mathematics Open Day 2021: Dif...  

Integrating factors explained:    • Oxford Calculus: Integrating Factors ...  

Solving simple PDEs:    • Oxford Calculus: Solving Simple PDEs  

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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-c...

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