I watched this video by 3Blue1Brown on how to find zeroes of a continuous map from $\Bbb R^2$ to $\Bbb R^2$: https://www.youtube.com/watch?v=b7FxPsqfkOY
He generalises intermediate value theorem on $\Bbb R$ to the plane to become "winding number not zero implies there is a zero of the function", and bisection method to bisecting rectangles, using the fact that winding numbers of loops add together nicely.
I think this generalisation can be done to any higher dimensions. Here, I try to formulate the generalisation:
Suppose $U\subseteq\Bbb R^{n+1}$ is open, convex and nonempty. Let $f:U\to\Bbb R^{n+1}$ be continuous. Suppose $S\subseteq U$ is a set that is homeomorphic to the $n$-sphere and for all $x\in S$, $f(x)\not=0$. We restrict $f$ to $S$ and obtain the function $g:S\to\Bbb S^n, g(x)=\frac{f(x)}{\lVert f(x)\rVert}$, where $\Bbb S^n$ is the $n$-sphere. We can calculate the degree of this map. I claim without proof the following: if the degree of $g$ is nonzero, then there is a zero of $f$ in the region bounded by $S$.
The generalisation of bisection method would be as follows:
We can further subdivide the region bounded by $S$ into two regions $R_1,R_2$, such that their boundaries are $S_1,S_2$ respectively, and $S_1+S_2=S$ (for the part where $S_1,S_2$ intersects each other, their "sum" would "cancel out each other" as they traverse in opposite orientations). Suppose $f\not=0$ on $S_1\cap S_2$. Then we can calculate the degrees of $g_1:S_1\to\Bbb S^n, g_2:S_2\to\Bbb S^n, g_1(x)=\frac{f(x)}{\lVert f(x)\rVert}, g_2(y)=\frac{f(y)}{\lVert f(y)\rVert}$. I claim without proof the following: the degree of $g$ is the sum of degrees of $g_1$ and $g_2$. As a result, when $g$ has nonzero degree, one of $g_1$ and $g_2$ must have nonzero degree and a zero of $f$ must be located in one of $R_1$ and $R_2$.
Are the above generalisations true statements? If so, where can I find proofs of the above?