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Unformatted text preview: 1.1. SUBMANIFOLDS OF EUCLIDEAN SPACE 3 be a subset of the Euclidean space R m described by the locus of r equations F ( x 1 , . . . , x m ) = y , = 1 , . . . , r . Suppose that M is non-empty and let x M , that is F ( x ) = y . Then the Implicit Function Theorem says that if det F x ( x ) ! , , where = 1 , . . . , r and = n + 1 , . . . , n + r , then there is a neighborhood of x such that the last r coordinates can be expressed as smooth functions of the first n coordinates: x = f ( x 1 , . . . , x n ) , = n + 1 , . . . , n + r . If this is true for any point of M , then M is a n-dimensional submanifold of R m . The matrix F x j ! , where = 1 , . . . , r and j = 1 , . . . , m is called the Jacobian matrix . The General Implicit Function Theorem says that if at a point x the Ja- cobian matrix has the maximal rank equal to r , rank F x j ( x ) !...
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