I'm studying multivariable calculus and I've encountered Stokes' theorem: ∮_(∂ S) F · dr = ∬_S (\ abla \ imes F) · n dS I'm told this is a generalisation of the fundamental theorem of calculus, but I don't see the connection. The original FTC says: ∫_a^b f'(x) dx = f(b) - f(a) How is Stokes' theorem related to this? And how does it relate to the divergence theorem and Green's theorem? Can someone explain the \"big picture\" of the fundamental theorems of vector calculus? Also, what is a concrete application of Stokes' theorem?

I am studying abstract algebra and I need to understand the structure of subgroups of cyclic groups. The theorem: Every subgroup of a cyclic group is cyclic. Moreover, for G = ⟨ g ⟩ of order n, for each divisor d of n, there is exactly one subgroup of order d, generated by g^(n/d). My questions: 1. How do I find all subgroups of ℤ_(12) (additive group modulo 12)? 2. How do I draw the subgroup lattice diagram? 3. What changes if G is infinite cyclic (ℤ)? 4. How does this theorem help in understanding Euler's totient function φ(d) — the number of generators of a cyclic group of order d? 5. How many elements of order 6 are there in ℤ₆ × ℤ₆? I want to see the complete subgroup structure.

Euclid's proof that there are infinitely many primes is famous, but I want to understand it deeply. Assume p₁, p₂, \ldots, pₙ are all the primes. Consider N = p₁ p₂ ⋯ pₙ + 1. Then N is either prime or has a prime factor not in our list, contradiction. My question: Could N itself be divisible by one of the existing primes? Euclid says no because dividing N by any pᵢ leaves remainder 1. But is that rigorous enough? And are there other proofs of infinitude of primes?

I'm studying geometry and I learned that for any convex polyhedron: V - E + F = 2 where V is vertices, E is edges, and F is faces. This is Euler's formula. I have verified it for cubes, tetrahedra, octahedra, etc. But does this formula hold for all polyhedra? What about non-convex ones? And what is the deeper topological meaning of the number 2? I've heard about the Euler characteristic χ = V - E + F generalising to other surfaces. How does that work?