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Degree Theory: Old and New

  • Haïm Brezis
Conference paper
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 27)

Abstract

The theory of degree has a long history which is a cascade of successive generalizations. Presumably, the oldest notion is the degree of a (smooth) map u from S 1 into S 1 (S 1 = the unit circle). The degree of u, also called winding number, counts “how many times u covers its range taking into account the algebraic multiplicity.” More generally, a smooth (say C 1) map u from S 1 into ℂ, such that u ≠ 0 on S 1 has a degree which may be computed through the very classical integral formula
$$\deg u = \frac{1} {{2\pi i}}\int_{S^1 } {\frac{{\dot u}} {u}}$$
(1.1)
which measures the “algebraic change of phase” of u as the variable goes around S 1 once. Similarly, if Γ is a simple curve in ℝ2 and u is a smooth map from Γ into S 1, then its degree can be computed as
$$\deg (u,\Gamma ) = \frac{1} {{2\pi }}\int_\Gamma {u \times u_\tau ,}$$
(1.2)
where × denotes the cross product of vectors in ℝ2 (here S 1 is viewed as a subset of ℝ2, not ℂ)and u τ denotes the tangential derivative of u along Γ.

Keywords

Toeplitz Operator Degree Theory Sobolev Class Algebraic Multiplicity Fractional Sobolev Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Boston 1997

Authors and Affiliations

  • Haïm Brezis
    • 1
    • 2
  1. 1.Analyse NumériqueUniversité P. et M. CurieParis CedexFrance
  2. 2.Department of MathematicsRutgers UniversityNew BrunswickUSA

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