We all know that the interior angles of a triangle sum to 180^°, but what is the formal proof? I know one proof uses the fact that alternate interior angles are equal when a transversal cuts parallel lines, but are there other approaches? Also, does this hold in non-Euclidean geometry? What is the sum of angles in a triangle on a sphere?

I'm studying differential geometry and I need to compute the Gaussian curvature of a parametric surface r(u, v). Given the first fundamental form: E = r_u · r_u, F = r_u · r_v, G = r_v · r_v And the second fundamental form: L = r_(uu) · n, M = r_(uv) · n, N = r_(vv) · n The Gaussian curvature is: K = LN - M²/EG - F² Can someone walk through this computation for a specific surface like a torus or a helicoid?

I know how to compute determinants, but I want to understand what they mean geometrically. For a 2 × 2 matrix, the absolute value of the determinant gives the area of the parallelogram formed by its column vectors. For a 3 × 3 matrix, it gives the volume of the parallelepiped. But what about: 1. The sign of the determinant indicating orientation 2. Why a zero determinant means the matrix is singular (non-invertible) 3. The determinant of a linear transformation being the factor by which volumes scale Can someone explain with diagrams or concrete examples?

I'm starting to learn about manifolds. The definition is: A topological space M is an n-dimensional topological manifold if: 1. M is Hausdorff 2. M is second-countable 3. M is locally Euclidean: each point has a neighborhood homeomorphic to an open subset of ℝⁿ The homeomorphisms φ: U → φ(U) ⊆ ℝⁿ are called coordinate charts. Why do we need the Hausdorff and second-countable conditions? Can someone give an example of a locally Euclidean space that is NOT a manifold because it fails these conditions? And what role do charts play in defining calculus on manifolds?

I am studying graph theory and I want to understand planar graphs — graphs that can be drawn on a plane without edge crossings. I know Kuratowski's theorem: A graph is planar iff it contains no subdivision of K₅ (complete graph on 5 vertices) or K_(3,3) (complete bipartite graph on 3+3 vertices). But: 1. What does a "subdivision" mean exactly? 2. Why are K₅ and K_(3,3) non-planar? 3. How do I check if a given graph contains a K₅ or K_(3,3) subdivision? 4. What is Euler's formula V - E + F = 2 for planar graphs and how is it used? Also, do all planar graphs have a straight-line drawing?