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)


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
$$\deg u = \frac{1} {{2\pi i}}\int_{S^1 } {\frac{{\dot u}} {u}}$$
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
$$\deg (u,\Gamma ) = \frac{1} {{2\pi }}\int_\Gamma {u \times u_\tau ,}$$
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.


Vortex Manifold Convolution Stein 


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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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