By M. Morse

Morse conception is a examine of deep connections among research and topology. In its classical shape, it presents a dating among the severe issues of yes soft features on a manifold and the topology of the manifold. it's been utilized by geometers, topologists, physicists, and others as a remarkably powerful software to review manifolds. within the Eighties and Nineteen Nineties, Morse thought was once prolonged to endless dimensions with nice luck. This publication is Morse's personal exposition of his rules. it's been referred to as the most vital and influential mathematical works of the 20th century. Calculus of diversifications within the huge is unquestionably one of many crucial references on Morse idea.

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**The calculus of variations in the large**

Morse thought is a research of deep connections among research and topology. In its classical shape, it offers a dating among the serious issues of convinced gentle capabilities on a manifold and the topology of the manifold. it's been utilized by geometers, topologists, physicists, and others as a remarkably potent instrument to review manifolds.

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We can put some of the results above (as well as some of the results we encounter in their particular cases below) into a broader context as follows. A point set A is called convex if A ∈ A & B ∈ A implies (AB) ⊂ A for all points A, BA. 16. Consider a ray OA , a point B ∈ OA , and a convex set A of points of the line aOA . If B ∈ A but O∈ / A then A ⊂ OA . 51 c Proof. Suppose that there exists C ∈ OA ∩ A. Then O ∈ A in view of convexity, contrary to hypothesis. Since c A ⊂ Pa and OA ∩ A = ∅, O ∈ / A, we conclude that A ⊂ OA .

1. If points C, D lie respectively on the sides h = OA and k = OB of the angle ∠(h, k) then ∠COD = ∠(h, k). Proof. (See Fig. 3. 2. Given an angle ∠AOB, we have B ∈ / aOA , A ∈ / aOB , and the points A, O, B are not collinear. 3 c c , Proof. Otherwise, we would have B ∈ aOA & B = O =⇒ B ∈ OA ∨ B ∈ OA =⇒ OB = OA ∨ OB = OA contrary to hypothesis that OA , OB form an angle. 3 ¬∃b (A ∈ b & O ∈ b & B ∈ b) and A ∈ / aOB . ✷ 71 In practice the letter used to denote the vertex of an angle is usually omitted from its ray-pair notation, so we can write simply ∠(h, k) 72 Thus, the angle ∠AOB exists if and only if the points A, O, B do not colline.

2, B ∈ aOA contradicts the fact that the rays OA , OB form an angle. 2. In the future the reader will encounter many such analogies. 88 Obviously, this means that given an angle ∠(h, k), none of the interior points of an angle ∠(k, m) adjacent to it, lies inside ∠(h, k). 15 is applied here to every point of the ray O . C 90 If O ⊂ Int∠AOB we have nothing more to prove. 47: Given an angle ∠(h, k), all points inside any angle ∠(k, m) adjacent to it, lie outside ∠(h, k). 48: If points B, C lie on one side of aOA , and OB = OC , either OB lies inside ∠AOC, or OC lies inside ∠AOB.