Ngô Quốc Anh

July 15, 2010

The winding number is a special case of the Brouwer degree

Filed under: Giải tích 7 (MA4247) — Ngô Quốc Anh @ 18:25

In the classical complex analysis, the winding number of a plane curve \Gamma with respect to a point a \notin \mathbb C \setminus \Gamma is defined by

\displaystyle w(\Gamma ,a) = \frac{1}{{2\pi i}}\int_\Gamma {\frac{{dz}}{{z - a}}} .

Let \Omega \subset \mathbb R^n be open and bounded. Let f : \Omega \to \mathbb R^n and p \notin f(\partial \Omega). Notation J_f(a) denotes the Jacobian of f evaluated at a. It is well-known that the Brouwer Degree Theory, usually denoted by \deg, is constructed for continuous function, C^0(\overline\Omega)-class, via the following steps

For the C^1(\overline\Omega)-class: We assume J_f(f^{-1}(p)) \ne 0 then

\displaystyle\deg (f,\Omega ,p) = \begin{cases} \sum\limits_{x \in {f^{ - 1}}(p)} {{\rm sgn} {J_f}(x)} , & {f^{ - 1}}(p) \ne \emptyset ,\\ 0, & {f^{ - 1}}(p) = \emptyset.\end{cases}

For the C^2(\overline\Omega)-class: In case we want to remove the condition J_f(f^{-1}(p)) \ne 0, we then define

\displaystyle\deg (f,\Omega ,p) = \deg (f,\Omega ,p')

where p' is any regular value of f sufficiently closed to p in the sense that

\displaystyle\left\| {p - p'} \right\| < {\rm dist}(p,f(\partial \Omega )).

For the C^0(\overline\Omega)-class: In this case, we define

\displaystyle\deg (f,\Omega ,p) = \deg (g,\Omega ,p)

where g \in C^2(\overline\Omega) is sufficiently closed to f in the sense that

\displaystyle\left\| f-g \right\| < {\rm dist}(p,f(\partial \Omega )).

Now we prove the following fact

Theorem.  Let B(0,1)\subset \mathbb C be the unit ball and \Gamma = \partial B(0,1). Assume f : \overline{B(0,1)} \to \mathbb C is a C^1 function and a \notin f(\Gamma). Then

\displaystyle\deg (f,B(0,1),a) = \frac{1}{{2\pi i}}\int_{f(\Gamma )} {\frac{{dz}}{{z - a}}} .

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