Search results for “формула Стокса”

Integration with respect to arc length: parametrising a curve, the length element $ds = |\mathbf{r}'(t)|\,dt$, arc length, curves in polar coordinates, the mass and centre of mass of a wire, symmetry instead of computation.

Finite geometries — the Platonic solids. A reminder — right actions, orbits, stabilisers, fundamental domains (corrected definition) and the example of $\mathbb{Z} + \mathbb{Z}$ on the plane; the formula $|G| = |\operatorname{Orb}(x)| \cdot |\operatorname{Stab}(x)|$ and the 48 symmetries of the cube; regular polyhedra — Sosinsky's definition and the definition via flags, a stellated counterexample; the five Platonic solids; regular polytopes in $\mathbb{R}^4$ (six) and in $\mathbb{R}^n$ for $n \ge 5$ (the cube, the simplex, the hyperoctahedron) with the coordinates of their vertices; why there are no more than five — face angles and the Schläfli symbol; uniqueness of the gluing and the rigidity theorem; Euler's formula; the finite subgroups of $SO(3)$ — $\mathbb{Z}_n$, $D_n$ and the rotation groups of the tetrahedron, cube and icosahedron; the duality cube–octahedron, icosahedron–dodecahedron; which figures on the sphere realise $\mathbb{Z}_n$, $D_n$ and $\mathbb{Z}_2$.

What algebra studies — algebraic structures. Groups (definition, examples, additive notation, uniqueness of the identity and of inverses), rings (variants of the axioms, rings with and without identity), fields and division rings, vector spaces and modules, algebras over a field. The symmetric group $S_n$ — permutations, non-commutativity, two-row notation, cycles and transpositions, decomposition into transpositions, disjoint permutations commute, decomposition into disjoint cycles and its uniqueness, powers, inversions, parity and the sign of a permutation, multiplying by a transposition changes the sign.