Jacobian Geometry

Multivariate calculus from first principles

Make the Jacobian square (n inputs, n outputs) and its determinant takes on a concrete geometric job. From Linear Algebra, the determinant of a matrix is the factor by which it scales volume. The Jacobian determinant tells you how much a map stretches or shrinks a tiny patch of space as it passes through.

If |det J| > 1, a little box of input space comes out bigger, so the map expands. If |det J| , it comes out smaller, so the map contracts. If det J = 0, the box gets squashed flat: the map collapses a dimension and is locally non-invertible.

Draw a tiny square on a sheet of stretchy rubber, then pull the sheet to distort the grid. The Jacobian determinant is the single number telling you how much that little square's area grew or shrank in the stretch. Pull the rubber both ways and the square balloons; squash it onto a single crease and its area drops to zero.

Where this lives in MLSuppose you want to bend a plain Gaussian into a complicated data distribution. A normalizing flow does exactly that, learning an invertible map g from simple density to complex one. As g stretches space, probability mass would leak unless you rescale it, so the change-of-variables formula p_X(x) = p_Z(g⁻¹(x))·|det J| uses the Jacobian determinant to keep total probability equal to 1. Flow…

▶ Jacobian Geometry

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