{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

differential geometry w notes from teacher_Part_2

differential geometry w notes from teacher_Part_2 - 3 1.1...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
1.1. SUBMANIFOLDS OF EUCLIDEAN SPACE 3 be a subset of the Euclidean space R m described by the locus of r equations F α ( x 1 , . . . , x m ) = y α 0 , α = 1 , . . . , r . Suppose that M is non-empty and let x 0 M , that is F ( x 0 ) = y 0 . Then the Implicit Function Theorem says that if det F α x β ( x 0 ) 0 , where α = 1 , . . . , r and β = n + 1 , . . . , n + r , then there is a neighborhood of x 0 such that the last r coordinates can be expressed as smooth functions of the first n coordinates: x α = f α ( x 1 , . . . , x n ) , α = 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 α x j , where α = 1 , . . . , r and j = 1 , . . . , m is called the Jacobian matrix . The General Implicit Function Theorem says that if at a point x 0 the Ja- cobian matrix has the maximal rank equal to r , rank F α x j ( x 0 ) = r , then there exists a coordinate system in a neighborhood of x 0 such that the last r coordinates can be expressed as smooth functions of the first n coor- dinates. If this is true for every point of M , then M is a n -dimensional submanifold of R m . The number r is called the codimension of M . If the codimension r is equal to 1, that is n = m - 1, then M is called a hypersurface . di ff geom.tex; April 12, 2006; 17:59; p. 7
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern