Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 349such that the end point of γ j , (j = 1, . . . , n) is equal to the beginning pointof γ j+1 . If γ j is defined on the interval [a j , b j ], this means thatγ j (b j ) = γ j+1 (a j+1 ).We call γ 1 (a 1 ) the beginning point of γ j , and γ n (b n ) the end point of γ j .The path is said to lie in an open set U ⊂ M C if each curve γ j lies in U,i.e., for each t, the point γ j (t) lies in U.An open set U is connected if given two points α and β in U, thereexists a path γ = γ 1 , γ 2 , . . . , γ n in U such that α is the beginning pointof γ 1 and β is the end point of γ n ; in other words, if there is a path γ inU which joins α to β. If U is a connected open set and f a holomorphicfunction on U such that f ′ = 0, then f is a constant. If g is a function onU such that f ′ = g, then f is called a primitive of g on U. Primitives canbe either find out by integration or written down directly.Let f be a C 0 −function on an open set U, and suppose that γ is a curvein U, meaning that all values γ(t) lie in U for a ≤ t ≤ b. The integral of falong γ is defined as∫γ∫f =γf(z) =∫ baf(γ(t)) ˙γ(t) dt.For example, let f(z) = 1/z, and γ(θ) = e iθ . Then ˙γ(θ) = ie iθ . Wewant to find the value of the integral of f over the circle, ∫ dz/z, so 0 ≤ θ ≤γ2π. By definition, this integral is equal to ∫ 2πie iθ /e iθ dθ = i ∫ 2πdθ = 2πi.0 0∫The length L(γ) is defined to be the integral of the speed, L(γ) =b| ˙γ(t)| dt.aIf γ = γ 1 , γ 2 , . . . , γ n is a path, then the integral of a C 0 −function f onan open set U is defined as ∫ γ f = ∑ n∫i=1f, i.e., the sum of the integralsγ iof f over each curve γ i (i = 1, . . . , n of the path γ. The length of a path isdefined as L(γ) = ∑ ni=1 L(γ i).Let f be continuous on an open set U ⊂ M C , and suppose that f hasa primitive g, that is, g is holomorphic and g ′ = f. Let α, β be two pointsin U, and let γ be a path in U joining α to β. Then ∫ f = g(β) − g(α);γthis integral is independent of the path and depends only on the beginningand end point of the path.A closed path is a path whose beginning point is equal to its end point.If f is a C 0 −function on an open set U ⊂ M C admitting a holomorphicprimitive g, and γ is any closed path in U, then ∫ γ f = 0.Let γ, η be two paths defined over the same interval [a, b] in an open set