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Unformatted text preview: Math 481
5. Implicit Function Theorem 1. Solutions of Constraint Equations
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a) Simplest Case: M: {($1,“ ,3:”+1) E Rn+1::c”+l —.~ h(3:1,. ,sc”+1)}.
(M: graph of h ) M 18 a manifold, with an atlas consisting of one coordinate patch. b) Now let M: “3:1,. ,x’H‘I) E Rn“: '1g(3: ,... ,$”+1)=0}. Suppose Dy # (0,”
at every p E M. Then, the Implicit Function Theorem states that (i) M is an n—dimensional manifold (ii) Around any 3? E M, there Is a coordinate patch ( W, 35: W —) R”) where Q5 18 projection onto an open set g'W 1n the (3:1,. ..,3:’5, . .. ,3:"'+1) plane.
Here, 3: is any coordinate such that 63‘ —g(p) # 0.
(iii) On W, 3:. 0 qt” is a smooth function h(3:,. ,zii, . .. ,xn‘l'l), so W' IS the graph of [2: W —> R (referred to as the‘ ‘implicit function”) 2. Example: M 2 unit sphere in Rn“.
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Simple guiding example: M = 51 C R2. 3. Implicit Function Theorem (General Form) : Assume : a) Consider the solution set of 1: equations in n + 19 variables:
For an open set U in Rn+k, let M: {oi1,... ,a:”+k) e U ; 91931,... an“) 20,... ,gk(x1,... ,x”+k) = 0} Le. g(wl,... ,wn‘l'k) = (g1(a:1,... ,m”+k),... ,gk(x1,... ,$”+k)) deﬁnes a map
9 : U—HR" and Mzg'1(0,... ,0).
b) Suppose that rank Dg = k at every y E M. This means that the rows of 81 81
139 == I I
QL" 63" 3x1 '* 8xﬂ+k are linearly independent. This happens‘if and only if the only solution of the equation 61(I‘OW1) + + ck(rowk) = 0 is c1 = 2: Ck = 0. Equivalently, for some choice of k columns of By, the
corresponding k X k subdeterminant is nonzero. We say that “ g is a. submersion at
every 3; E M ”. Conclusion : a) M is an ndimensional manifold. 13) At any y E M , you can get a chart around y as follows: Take any choice of k
columns , say columns 31, . .. ,jk, of Dg(y), so that the k x k: matrix they form has
nonzero determinant. Then there is a chart (W,¢) around y E M, where gt : W —> R”, is projec—
tion onto an open set W in the coordinate plane of the remaining n variables 1 A‘ h' k
(:c ,... ,m“,... 3“,... ,x“+ ).
c) On W, the implicit functions 5:31 ego—1,. .. ,ka 0451 are smooth functions W —> R: $j10¢_1 2 hj1($1,...,Ej1,...,ﬁj",...,$"+k) mikes"1 that”,ei1,...,sik,...,xn+k) ...
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 Fall '08
 kapovich
 Math

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