Introduction to the Complex Analysis of Minimal Surfaces

Let c : I → C and f : C → R. To define the differential df|p : C → R of the function f choose a tangent vector v ∈ C at p, pick a curve c such that c(0) = p, c′(0) = v and set: df|p · v := df|p(v) := (f ◦ c)′(0). The differentials dx|p, dy|p of the functions x(z) := Re (z) and y(z) := Im (z) are not depending on the base point p. They are always denoted dx, dy and they are taken as basis for representing other differentials or differential forms ( = linear combinations of differentials, functions as coefficients). In C one uses always the standard basis ex = 1, ey = i and therefore writes: v = vx · 1 + vy · i, df|p(v) = vxdf|p(1) + vydf|p(i) = vx ∂ ∂x f|p + vy ∂ ∂y f|p,