... The concept of regular value and the theorem of Sard and Brown, which asserts that every smooth mapping has regular values, play a central role. ... Smooth manifolds and smooth maps Tangent spaces and derivatives Regular values The fundamental theorem of algebra v11 1 2 7 8 2. ... Proof of Sard's theorem 4. ... For any smooth map f : U + V the derivatiue dfz ; Rk+ R' is defined by the formula df,(h) = Sometimes we will need to look at manifolds of dimension zero. ... CONSIDER a smooth map f : S" + S...