In differential geometry, Stokes' theorem (also called the generalized Stokes' theorem) is a statement about the integration of differential forms on manifolds, which generalizes several theorems from vector calculus. William Thomson first discovered the result and communicated it to George Stokes in ...
the statement of the theorem is very brief and succinct, and itsproof presents little difficulty.[...
This is the statement of Stokes' theorem. It relates the circulation of the vector field F around the boundary C of the surface S to the surface integral of the curl (∇ × F) over the surface S. The proof of Stokes' theorem relies on the concept of the curl and the application of...
statement of the theorem is by William Thomson (Lord Kelvin) and appears in a letter of his to Stokes. The theorem acquired its name from Stokes' habit of including it in the Cambridge prize examinations. Let M be an oriented piecewise smooth manifold of dimension n and let ω be an n...
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1.1 Statement of Results Our first main Theorem addresses (Q1). Just as Calderón [16] filled the supercritical gap to find solutions corresponding to initial data in [Math Processing Error]Lp for [Math Processing Error]2<p<3, between Leray ([Math Processing Error]L2(R3)) and Kato’s mild...
in order to solve this i thought of using the stokes theorem because the normal to the plane is \[12(0,1,1)\] thus giving me ∮Fdr=∫∫curl(F)∗n∗ds=∫∫2/sqrt2∗sin(xyz) i tried to parametries x y and z x= rcos(t)+1 y=rsin(t)+2 z=1/2-rsin(t) but ...
PDEs and other complex-valued SDEs, using basic concepts from complex geometry. Our main result is to verify this condition for the Galerkin truncations of the 2d Navier–Stokes equations with frequency cutoffon torii of any aspect ratio (Theorem1.1), thus proving chaos for allsufficiently small...
-maximal regularity, see theorem 3.5 . in sect. 4 , we prove that all equilibria of ( 1.5 ) are stable. moreover, we show that any solution starting close to an equilibrium exists globally and converges at an exponential rate to a (possibly different) equilibrium as time tends to...
Theorem 1.1 Let Ω ⊂ ℝ3 be a bounded domain with smooth boundary, and suppose that ρ≥ 0,μ > 0 and (1.6) α>1, and that Λ∈ W2,∞(Ω). Then for any choice of (n0, c0, u0) fulfilling (1.5), the problem (1.4) possesses at least one global generalized solution (n,...