What is an intuitive explanation of Jacobians and a change of basis?

Later you move on to doing geometry on surfaces and you might find yourself wanting to change coordinates on a surface. Now there's no global coordinate system that will have the same matrix everywhere – the transformation matrix between two coordinate systems will depend on where on the shape you are. That's the Jacobian, sort of.  In 3d a surface has two dimensions so you usually see 2×2 matrices for the Jacobian. The vector spaces on a surface are the "tangent spaces",  the spaces of possible vectors tangent to the surface at that point, and a basis is a choice of linearly-independent tangent vectors that can be combined to make any tangent to a curve at that point.

Answer by Alex Kritchevsky:

Change of basis:
You're close with both your pictures. I prefer the first one.

The perspective we like to take in physics is: here's the universe. It has (shapes/vectors/lines/worldlines/whatever) on it. They have a real shape, before you ever put coordinates anywhere.

Say you pick rectangular coordinates – an origin, three axes, and units for each one (they don't have to be the same – you can skew things as you like). Every point and shape gets coordinates the describe it in this.

Pick new coordinates. Maybe they're rotated, maybe they're stretched really far in one direction, maybe they're spherical, maybe they're just totally. As long as they still have three dimensions they can still describe every point. If your three weird directions are in a plane they can't so they're not a 'basis' for the 3d space.

If you have two sets of coordinates the change of basis matrix translates coordinates in one to coordinates in another. It may do all sorts of crazy skewing, rotating stuff.. but it will be invertible (it's just a perspective shift). That's equivalent to: it can't be a projection (or it would 'lose' a dimension); it can't have a zero determinant (it has to have three linearly independent rows/columns – the transforms in both directions); etc. If it has a negative determinant it'll invert your coordinates so things get negative (signed) volumes where they had positive ones; that is, the 'handedness' or "parity" of the shapes will switch (if that means anything to you).

Note that change of basis matrices do not move the origin. Well, normally – though if you get into projective geometry you can do it.

Jacobians:

Later you move on to doing geometry on surfaces and you might find yourself wanting to change coordinates on a surface. Now there's no global coordinate system that will have the same matrix everywhere – the transformation matrix between two coordinate systems will depend on where on the shape you are. That's the Jacobian, sort of.  In 3d a surface has two dimensions so you usually see 2×2 matrices for the Jacobian. The vector spaces on a surface are the "tangent spaces",  the spaces of possible vectors tangent to the surface at that point, and a basis is a choice of linearly-independent tangent vectors that can be combined to make any tangent to a curve at that point.

The first use of the Jacobian you're exposed to is typically in an integral as sort of a multivariable u-substitution (which is in fact exactly what it is). In [math] \int \int_{\sigma}dx dy = \int \int_{\sigma} r dr d\theta [/math], the 'r' is the Jacobian determinant. What's really going on here is: [math] dx dy [/math] is actually a thing called a wedge product – an infinitesimal "area vector" on the surface, basically a cross product of the tangent directions (well, technically, differential forms):
[math]dxdy=dx \wedge dy[/math]. When you do the change of basis each of 'dx' and 'dy' get a factor of the full Jacobian – but the way the wedge product works means that their [math]dx \wedge dy [/math] gets a factor of [math] det(J) [/math].

The intuitive version of that is: since [math]dxdy[/math] is an (infinitesimal) area, and each is transformed by one row of the Jacobian, then the cross product of the rows gives the scaling factor for the area – and the magnitude of the cross product of the rows of a 2×2 matrix is the Jacobian of the matrix (equivalently the triple product in 3d, etc).

What is an intuitive explanation of Jacobians and a change of basis?

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