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.

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.

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[shashwatkumar]

You're exactly like how Machine Gun Kelly would have looked if he taught Calculus

[abbe]

Feels great to know why the Jacobian comes into the calculations when switching coordinate systems. I never learned that while doing multivariate calculus this past semester. Keep up the good work! Regards from a fellow math nerd from Sweden.

[dmc-au]

I brushed across the Jacobian while learning statistics recently. It seemed reasonable that we'd need to scale by the change of space in that context, but this video made it concrete as to what was going on behind the scenes. Thanks, Tom!

[megistone]

I'm shocked how you've packed many topics such as vector product, Jacobian, areas, and more into such a video, while clearly explaining Jacobian, the main topic. Even if I don't speak English well I can understand it and it is very interesting to watch the explanation and behavior as if you are transmitting energy to the viewer. I'm very satisfied.

[leocheung]

Hey there! The second you explained the Jacobian as the stretch factor of converting from one coordinate system to another, I understood it so much better! This was so much better of an explanation than my textbook

[davidgillies]

This really should be taught at A-level rather than first-year undergrad courses. Jacobians act as a nice sliproad onto the main highway of tensors and differential geometry in general, whose introduction is in turn often delayed (or even omitted) at bachelor's level.

[paulmcghee]

Thanks for your exceptional work Tom. I've got a degree in maths and still learning little things like this really makes sure I keep lifting my knowledge.

You're putting a load of effort into these videos. It is greatly appreciated.

[Deviated]

I envy the ability to be good and understand math, I’m doing intermediate algebra right now in college and I’m having a hard time grasping the concept. Love your videos, keep it up!

[sarkiskaralian]

You sir are a very valuable math resource for students and perhaps even teachers. Thank you!

[LucasVieira-obfx]

This is by far the most comprehensible explanation of the Jacobian I've ever found. Nice work!

[saraswatasensarma]

Thank you so much. I am a first year Maths student from India, and these simple yet beautiful concepts are what keep mathematics in my heart. Keep up the great work Sir!!

[AcryllixGD]

This is such a fantastic video! I'm currently in year 13, thinking of doing a maths degree, im fascinated with calculus, its by far my favourite aspect of maths, not only did multivariable integration make sense but also the use of determinants. Amazing video!

[jpmagnos]

You really are saving me in university... I feel like I can understand where things comes from and why they are the way they are when you explain it... much better than my university professor who is more interested in making us fail class

[daniellund]

Absolutely love this video, currently in the process of studying vector calculus (and some other stuff I also don't understand) for machine learning and struggled to wrap my head around jacobian's, this makes so much more sense now

[TomiThemself]

Tom never fails to explain what seem as hard mathematical concepts, in really easy way. Thank You!

[likithstochastic]

Thanks! That was explained in an intuitive way. I guess the key here is to think of the elemental rectangular areas changing in to rotated parallelograms during the coordinate transformation. The example you gave in the beginning with regard to the area of the circle makes the concept clearer.

[prille]

I love your explainations, I now have a better understanding of what I’ve learned in the past 😊 thanks so much for your videos

[kevindegryse]

I already knew how to use change of coordinates and Jacobian. But it is actually the first time I understand the geometric meaning of it :)
Thank you

[science-therealfun]

This video is too good. So informative and he explained such a difficult calculation so easily. Hats off and keep it up.Thanks Tom👍❤

[dv]

What a mesmerizing presentation. I had math through differential equations at university thirty-five years ago. If you had given lectures, such as you present here, perhaps the 4.0 GPA achieved would had met something. Grade Inflation was in full bloom. Thank you.