For each of the following examples, find two distinct triangulations. Then directly compute (using the triangulations) that both triangulations yield the same Euler characteristic.

1. A compact subset \(K\subset M\subset\mathbb{R}^3\) of a surface, such that \(K\) is homeomorphic to a closed disk.

2. The sphere \(S^2\subset \mathbb{R}^3.\) (Note: do not use the octant triangulation from the lecture notes.)

3. The torus \(T\subset \mathbb{R}^3.\)

Let \(M\subset\mathbb{R}^3\) be a compact embedded surface with non-positive Euler characteristic, i.e \(\chi(M)\le 0.\) Prove that \(M\) contains elliptic points, hyperbolic points and points such that the Gaussian curvature is zero.

Let \(\mathcal{X}:= [a,b]\times [c,d]\subset \mathbb{R}^2.\) Consider a \(1\)-form \(\alpha=f(x,y)dx+g(x,y)dy\) on \(\mathcal{X}\) such that \(d\alpha=0,\) where \(f,g\in C^{\infty}(\mathcal{X},\mathbb{R}).\) Show that there exists a function \(h\colon \mathcal{X}\rightarrow \mathbb{R}\) such that \(\alpha=dh.\)