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Closed and Exact Differential Forms CLOSED AND EXACT DIFFERENTIAL FORMS IN Rn PATRICIA R. CIRILO y, JOSE¶ REGIS A.V. FILHO z, SHARON M. LUTZ ? [email protected] , [email protected], [email protected] Abstract. We show in this paper that if every closed 1-form de¯ned on a domain of R2 is exact then such a domain is simply connected. We also show that this result does not hold in dimensions 3 and 4. 1. Introduction It is well known that every closed 1-form de¯ned on a simply connected domain ­ of Rn is exact (see Section 2 for the de¯nitions). This statement poses the following question: Suppose we have a domain ­ of Rn with the property that every 1-closed form is exact. Is this domains simply connected? We answer this question in the a±rmative for n = 2 and give counterexamples for n = 3; 4. The paper is organized as follows: Section 2 contains some de¯nitions and results that will be used in the later sections, section 3 proves the case for n = 2, section 4 de¯nes the Real Projective Plane that will be used to construct the counterexample for n = 4 and sections 5 and 6 present the counterexamples for cases n = 4 and n = 3 respectively. 2. Background: 1- Forms on Simply Connected Domains We start this section by recalling some de¯nitions of multivariable calculus. Let ­ be a domain of Rn, that is, a path connected open set of Rn. Let f : ­ ½ Rn ! R be a smooth function, that is, a function whose all partial derivatives of all orders are continuous. The gradient of f is de¯ned as @ @ r(f) = ( ; : : : ; ); @x1 @xn 1 ? 2 PATRICIA R. CIRILO y, JOSE¶ REGIS A.V. FILHO z, SHARON M. LUTZ and the directional derivative in the direction of a vector v = (v1; : : : ; vn) is given by @ @ df(v) = r(f) ¢ v = v1 + ¢ ¢ ¢ + vn: @x1 @xn Observe that if we denote by xi the projection map n xi : R ! R given by (x1; : : : ; xn) 7! xi, we have r(xi) = (0; : : : ; 1; : : : ; 0) ) dxi(v) = vi: It then follows that @f @f (1) df(v) = dx1(v) + ¢ ¢ ¢ + dxn(v): @x1 @xn n We know that for each point p 2 ­, df and the dxi's are linear maps from R to R, that is, they are elements of the dual space (Rn)¤. Therefore equation (1) shows that, as elements of (Rn)¤, df is written as the linear combination @f @f df = dx1 + ¢ ¢ ¢ + dxn: @x1 @xn This motivates the following de¯nition. De¯nition 2.1. A di®erential 1-form ! de¯ned on a domain ­ is a map that to each point p 2 ­ assigns !(p) 2 (Rn)¤ given by !(p) = a1(p)dx1 + ¢ ¢ ¢ + an(p)dxn; n such that each ai : ­ ½ R ! R is a smooth function. Example The 1-form y x ! = ¡ dx + dy x2 + y2 x2 + y2 de¯ned on ­ = R2 ¡ (0; 0). De¯nition 2.2. A di®erential 1-form ! de¯ned on a domain ­ is said to be closed if @a @a i (p) = j (p); 8i; j and 8x 2 ­: @xj @xi We say that a di®erential 1-form ! is exact if there exists a smooth function f : ­ ! R such that ! = df. n CLOSED AND EXACT DIFFERENTIAL FORMS IN R 3 Proposition 2.3. The following are equivalent: 1) ! is exact in a connected open set ­ 2) c ! depends only on the end points of c for all c ½ ­ 3) Rc ! = 0, for all closed curves c ½ ­ R This is a standard result on di®erential 1-forms, the reader can ¯nd the proof in several textbooks, for instance [1]. Recall that the Theorem of Schwarz states that if f is smooth then @2f @2f = : @xi@xj @xj@xi Since @f @f df = dx1 + ¢ ¢ ¢ + dxn; @x1 @xn that is, ai = @f=@xi, we conclude that exact implies closed. The converse of the last result is not true as we will show in the next example. Example 2.4. Let us consider the di®erential 1-form y x ! = ¡ dx + dy x2 + y2 x2 + y2 de¯ned before. We claim that ! is is closed but is not a exact 1-form. In fact, let γ be a closed curve such that γ : [0; 2¼] ¡! R2 θ 7! (cosθ; sinθ) Computing the line integral, y x ! = ¡ 2 2 dx + 2 2 dy Zγ Zγ x + y x + y 2¼ sinθ cosθ = ¡ 2 2 (¡sinθ)dt + 2 2 (cosθ)dt Z0 sinθ + cosθ sinθ + cosθ 2¼ = dt Z0 = 2¼ Since γ ! 6= 0, ! is not exact. R ? 4 PATRICIA R. CIRILO y, JOSE¶ REGIS A.V. FILHO z, SHARON M. LUTZ On oder hand, if we compute d! we have d! = dA ^ dx + dB ^ dy; where, y x A = ¡ and B = : x2 + y2 x2 + y2 @A @A @B @B d! = ( dx + dy) ^ dx + ( dx + dy) ^ dy @x @y @x @y @A @B = dy ^ dx + dx ^ dy @y @x @A @B = ¡ dx ^ dy + dx ^ dy @y @x @A @B = (¡ @y + @x )dx ^ dy = 0 Thus ! is a closed 1-form but is not exact. We note here that for domains of Rn with some special topological property, the converse is true, namely, closed implies exact. We explain next such a topological property. De¯nition 2.5. Let us consider the closed and bounded interval I = [0; 1]. A path in Rn is a continous map a : I ! Rn. The path is called a loop if it is closed, that is, a(0) = a(1) and if it does not intersect itself. We say that a loop a is homotopic to a point p if it is homotopic to the constant path c(t) = p for all t 2 [0; 1]. De¯nition 2.6. Two closed paths a; b : I ! X, with a(0) = a(1) = x0 2 X, are said to be homotopic (denoted by a ' b) if there is a continuous map H : I£I ! X such that H(t; 0) = a(t); H(t; 1) = b(t); H(0; s) = H(1; s) = x0 8s; t 2 I n CLOSED AND EXACT DIFFERENTIAL FORMS IN R 5 a(t) s ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢ H ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢ b(t) ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢ x ¡ ¡ ¡ ¡ I ¢¡¢¡¢¡¢¡¢ 0 ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢ 0 I t Intuitively, a homotopy is a continuous deformation that starts in a stage 0 (with a(t)) and ends in the stage 1 (with b(t)). During this deformation, the end points a(0) = b(0) = a(1) = b(1) remain ¯xed. Proposition 2.7. Let be a; b : I ! ­ ½ Rn. If a ' b and f : ­ ! Rn is a continuous function, then f(a) ' f(b). Lemma 2.8. The relation a ' b is an equivalence relation. The proof of this result can be found in all textbooks on the subject. We call the reader's attention that the main point is to show the transitivity property of the relation, namely, if a ' b and c ' d then ac ' bd, where ac is also a loop parametrized as follows a(2t) 0 · t · 1=2 ac(t) = ½ c(2t ¡ 1) 1=2 · t · 1 De¯nition 2.9. The fundamental group of a set X with respects to a basepoint x0, denoted by ¼1(X; x0), is the group formed by the set of equivalence classes of all loops, i.e., paths with initial and ¯nal points at the basepoint x0, under the equivalence relation of homotopy. Since ¼1(X; xo) is a group it has an identity element, which is the homotopy class of the constant path at the basepoint x0. If x0 and x1 are on the same connected by path component of X then the group ¼1(X; xo) is isomorphic to the group ¼1(X; x1). In other words, if ® 2 ¼1(X; x1) ¡1 there exists a correspondent element, that is γ®γ , in ¼1(X; xo). The intuitive ideia comes from the picture below. 3. The Theorem for n = 2 As mentioned in the introduction, for domains of R2 we have the following theorem. ? 6 PATRICIA R. CIRILO y, JOSE¶ REGIS A.V. FILHO z, SHARON M. LUTZ α = [a ] {γ = [c ] a c ¥¦ £¤ X −1 X o c 1 Theorem 3.1. If ­ is a subset of R2 such that any closed 1-form ! de¯ned in ­ is exact, then ­ is simply connected. The proof of theorem 3.1 is a consequence of two classical theorems for closed curves that are quoted below. Recall that a curve is called a simple curve if it does not intersect itself. Theorem 3.2. Jordan Curve Theorem: Let C ½ R2 be a closed simple curve. Then C divides the plane in two connected components: one bounded by C and the other unbounded. Theorem 3.3. (Schoenflies Theorem) If C is a closed simple curve in R2 then C bounds a region that is homeomorphic to a disk D = f(x; y) 2 R2jx2 + y2 · rg. Proof of the 3.1 - Suppose that the domain ­ is not simply connected. Then, by de¯nition, there exists a point p 2 ­ and a loop γ ½ ­ such that γ is not homotopic to p. Let J denote the interior of γ union with γ. It follows from Schoenflies theorem that J is homeomorphic to a disk in the plane. Therefore, γ is homotopic to any point p in J, since a disk of R2 is simply connected. It follows that there exists q in the interior of J such that q 62 ­. Without loss of generality, suppose that q = (0; 0) and consider ¡y x ! = dx + dy x2 + y2 x2 + y2 The 1-form ! is then a di®erential form on ­.
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