There is a part in the proof of sard's theorem where we restrict our discussion to a point x such that Df(x)=0, and then declare that f ' (x) is a proper (n-1) subspace (f is n-dim). What I don't understand is, the argument then goes by considering any two points in a sub-rectangle around this point and stating that all such points "lie within ε√n (l/N) of the (n-1)-plane V+f(x)." Where √n (l/N) is the length of the longest "diagonal" of our rectangles used in the proof and epsilon pops up from a "continuity" argument. Anyway, my question is about the V+f(x) plane.(adsbygoogle = window.adsbygoogle || []).push({});

I don't really see how its a plane. I believe its "centered" around f(x),.... is this correct?

Also, it says that {Df(x)(y-x): y in rectangle} lies in an (n-1)-dim subspace V of R^n. Is it saying that {Df(x)(y-x)} constitutes all of V ? or just that it is locally approximated by this vector space..... I feel like V is a plane tangent to this point, if that's the case then this all makes sense, but I'm not really sure..... All help appreciated. Are there any other suggestions on how I can "see" this.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Sard's theorem's vector space

**Physics Forums | Science Articles, Homework Help, Discussion**