## Sard’s theorem

I have recently been reading Smoller’s book ‘Shock Waves and Reaction-Diffusion Equations’ as background for a course on reaction-diffusion equations I am giving. In this context I came across the subject of Sard’s theorem. This is a result which I had more or less forgotten about although I was very familiar with it while writing my PhD thesis more than thirty years ago. I read about it at that time in Hirsch’s book ‘Differential Topology’, which was an important reference for my thesis. Now I had the idea that this might be something which could be useful for my present research, without having an explicit application in mind. It is a technique which has a different flavour from those I usually apply. The theorem concerns a (sufficiently) smooth mapping between $n$-dimensional manifolds. It is a local result so that it is a enough to concentrate on the case where the domain of the mapping is a suitable subset of Euclidean space and the range is the same space. We define a regular value of $f$ to be a point $y$ such that the derivative of $f$ is invertible at each point $x$ with $f(x)=y$. A singular value is a point of the range which is not a regular value. The statement of Sard’s theorem is that the set $Z$ of singular values has measure zero. By covering the domain with a countable family of cubes we can reduce the proof to the case of a cube. Next we write the cube as the union of $N^n$ cubes, by dividing each side of the original cube into $N$ equal parts. We need to estimate the contribution to the measure of $Z$  from each of the small cubes. Suppose that $y_0$ is a singular value, so that there is a point $x_0$ where the derivative of $f$ is not invertible with $f(x_0)=y_0$. Consider now the contribution to the measure of the image from the cube in which $x_0$ lies. On that cube $f$ can be approximated by its first order Taylor polynomial at $x_0$. The image is contained in the product of a subset of a hyperplane whose volume is of the order $N^{-(n-1)}$ and an interval whose length is of the order $\epsilon N^{-1}$ for an $\epsilon$ which we can choose as small as desired. Adding over the at most $N^n$ cubes which contribute gives a bound for the measure of the set of singular values of order $\epsilon$. Since $\epsilon$ was arbitrary this completes the proof. In words we can describe this argument as follows. The volume of the image of a region which intersects the set of singular points under a suitable linear mapping is small compared to the volume of the region itself and the volume of the image under the nonlinear mapping can be bounded by the corresponding quantity for the linear mapping up to an error which is small compared to the volume defined by the linear mapping.