= Outward vector field zero theorem
{title2=$g(x)\cdot x\geq0\text{ on }|x|=R\ \Longrightarrow\ \exists |x_0|\leq R:g(x_0)=0$}
A continuous <vector field> on a closed Euclidean ball whose boundary values have nonnegative outward component has a zero in the ball. Otherwise $T(x)=-Rg(x)/|g(x)|$ is a continuous self-map of the ball. Its <fixed point>, supplied by the <Brouwer fixed-point theorem>, lies on the boundary and satisfies $g(x)\cdot x=-R|g(x)|<0$, a contradiction. The argument turns a zero-finding problem into a <fixed point> problem without any differentiability assumption.
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