Brouwer degreeBrouwerDegree2004-12-10 13:03:092005-03-05 12:23:22Definition% this is the default PlanetMath preamble. as your knowledge
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\newtheorem*{defn}{Definition}Suppose that $M$ and $N$ are two oriented differentiable manifolds
of dimension $n$ (without boundary) with $M$ compact and $N$ connected and suppose that
$f \colon M \to N$ is a differentiable mapping. Let $Df(x)$ denote the
differential mapping at the point $x \in M$,
that is the linear mapping $Df(x) \colon T_x(M) \to T_{f(x)}(N)$. Let $\operatorname{sign} Df(x)$ denote the sign
of the determinant of $Df(x)$. That is the sign is positive if $f$ preserves
orientation and negative if $f$ reverses orientation.
\begin{defn}
Let $y \in N$ be a regular value, then we define the {\em Brower degree} (or just
degree) of $f$ by
\begin{equation*}
\operatorname{deg} f := \sum_{x \in f^{-1}(y)} \operatorname{sign} Df(x) .
\end{equation*}
\end{defn}
It can be shown that the degree does not depend on the regular value $y$ that we pick so that $\operatorname{deg} f$ is well defined.
Note that this degree coincides with the \PMlinkname{degree}{Degree5} as defined for maps of spheres.
\begin{thebibliography}{9}
\bibitem{Milnor:topdiff}
John~W. Milnor.
{\em \PMlinkescapetext{Topology From The Differentiable Viewpoint}}.
The University Press of Virginia, Charlottesville, Virginia, 1969.
\end{thebibliography}