In mathematics, an unfolding of a smooth real-valued function ƒ on a smooth manifold is a certain family of functions that includes ƒ.
Definition Let M {\displaystyle M} be a smooth manifold and consider a smooth mapping f : M → R . {\displaystyle f:M\to \mathbb {R} .} Let us assume that for given x 0 ∈ M {\displaystyle x_{0}\in M} and y 0 ∈ R {\displaystyle y_{0}\in \mathbb {R} } we have f ( x 0 ) = y 0 {\displaystyle f(x_{0})=y_{0}} . Let N {\displaystyle N} be a smooth k {\displaystyle k} -dimensional manifold, and consider the family of mappings (parameterised by N {\displaystyle N} ) given by F : M × N → R . {\displaystyle F:M\times N\to \mathbb {R} .} We say that F {\displaystyle F} is a k {\displaystyle k} -parameter unfolding of f {\displaystyle f} if F ( x , 0 ) = f ( x ) {\displaystyle F(x,0)=f(x)} for all x . {\displaystyle x.} In other words the functions f : M → R {\displaystyle f:M\to \mathbb {R} } and F : M × { 0 } → R {\displaystyle F:M\times \{0\}\to \mathbb {R} } are the same: the function f {\displaystyle f} is contained in, or is unfolded by, the family F . {\displaystyle F.}
Example Let f : R 2 → R {\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} } be given by f ( x , y ) = x 2 + y 5 . {\displaystyle f(x,y)=x^{2}+y^{5}.} An example of an unfolding of f {\displaystyle f} would be F : R 2 × R 3 → R {\displaystyle F:\mathbb {R} ^{2}\times \mathbb {R} ^{3}\to \mathbb {R} } given by
F ( ( x , y ) , ( a , b , c ) ) = x 2 + y 5 + a y + b y 2 + c y 3 . {\displaystyle F((x,y),(a,b,c))=x^{2}+y^{5}+ay+by^{2}+cy^{3}.}
As is the case with unfoldings, x {\displaystyle x} and y {\displaystyle y} are called variables, and a , {\displaystyle a,} b , {\displaystyle b,} and c {\displaystyle c} are called parameters, since they parameterise the unfolding.
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