The Jacobian, Geometrically (bollu.github.io)

I only learnt the (arguably) "real meaning" of the Jacobian once I studied differential geometry. I haven't seen this explained anywhere. I took a stab at an elementary exposition (with animations) of this view of the Jacobian as the object that tells us how to map tangents from one space to another!

This is very soothing.

Anyone who has Jacobian anxiety will surely be cured.