Differential Equations on the two dimensional torus 1 Constant ...

November 1, 2010 9b-62 Gradient vector fields on the torusTo specify a gradient vector field on the two dimensional torus, we need toconsider smooth potential functions on the torus T 2 = R 2 /Z 2 . These aresmooth functions ψ(x, y) such thatψ(x + 1, y + 1) = ψ(x, y) for all (x, y) ∈ R 2 .From the theory of Fourier series, we can write such functions as sums∞∑ψ(x, y) = a k cos(2πkx) + b k sin(2πk)k=0The simplest such examples are the trigonometric polynomialsN∑ψ N (x, y) = a k cos(2πkx) + b k sin(2πk)k=0for a positive integer N.Givena function ψ(x, y) with ψ(x + 1, y + 1) = ψ(x, y), we obtain itsgradient vector field byX(x, y) = (∂ x ψ(x, y), ∂ y ψ(x, y))We wish to give a simple geometric interpretation of this in a special case.First, we recall some concepts about surfaces of revolution in R 3 .Let (X, Y, Z) denote the coordinates of a point in R 3 .Let I = (a, b) be an open interval in R with a < b.Consider a C r parametrized curve η : I → R 3 in the XY -plane given bysuch thatandη(t) = (X(t), Y (t), 0)inf | X(t) |) > 0t∈Iinf |t∈I X′ (t) 2 + Y ′ (t) 2 | > 0