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# h5 - Math 481 5 Implicit Function Theorem 1 Solutions of...

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Unformatted text preview: Math 481 5. Implicit Function Theorem 1. Solutions of Constraint Equations ————____.________ 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“. -—-—-————._._._.__ 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: {oi-1,... ,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 n-dimensional 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 045-1 are smooth functions W —> R: \$j10¢_1 2 hj1(\$1,...,Ej1,...,ﬁj",...,\$"+k) mikes-"1 that”,ei1,...,sik,...,xn+k) ...
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h5 - Math 481 5 Implicit Function Theorem 1 Solutions of...

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