Multivariable Calculus 21 | Diffeomorphisms

Diffeomorphisms formalize when a change of coordinates is smooth in both directions—meaning a map has a smooth inverse, not just a smooth forward formula. That “invertibility with matching smoothness” becomes crucial later for results like the implicit function theorem, where one needs to guarantee that locally, variables can be traded without breaking differentiability.

To make this precise, the discussion uses the notation C^k for functions whose partial derivatives up to order k exist and are continuous. A map between open sets U ⊂ R^n and V ⊂ R^m is called a C^k diffeomorphism when three conditions hold: (1) the map is C^k, (2) it is bijective so an inverse exists, and (3) the inverse map is also C^k. In the special case k = ∞, the map is C^∞, which means it lies in every C^k class; equivalently, it has continuous partial derivatives of all orders.

A concrete example in two dimensions illustrates what “smoothness” means in practice. The map f: R^2 → R^2 given by f(x, y) = (x·y, x^2) has partial derivatives that remain continuous no matter which multi-index derivative is taken, so it qualifies as C^∞. The key point is that smoothness of the forward map alone doesn’t settle what happens after inversion.

The transcript then contrasts two one-dimensional functions that are both bijective on appropriate domains but behave differently when inverted. For f(x) = x^2 restricted to positive reals, the inverse is √x, which is differentiable and stays well-behaved; this yields a C^1 diffeomorphism, and in fact a C^∞ one. But for f(x) = x^3 on R, the inverse is the cube root. Even though f itself is smooth, the inverse has an infinite slope at 0: the derivative blows up, so the inverse fails to be C^1. This shows why the inverse’s differentiability cannot be ignored in the definition—smoothness of f does not guarantee smoothness of f^{-1}.

Next comes a necessary condition for a C^1 diffeomorphism using Jacobians. If f: U → V is C^1 and has a C^1 inverse, then composing f with f^{-1} gives the identity map on U and V. Differentiating these identities and applying the chain rule shows that the Jacobian matrices must multiply to the identity. Therefore, the Jacobian of f is invertible at every point in U, which implies its determinant is nonzero everywhere. In one dimension, this matches the familiar “derivative never vanishes” idea. However, the transcript emphasizes that while nonzero determinant is necessary, it is not sufficient in dimensions two and higher—higher-dimensional geometry allows more subtle failures than just a zero slope.