van Hieles 의 이론에 의한 국민학교 기하도형(幾何圖形)학습의 분석연구

서성보 한국수학교육학회 1995 수학교육 Vol.34 No.2

法曲率의 두개의 極値 : Sur les Extre^mes de la Courbure Normale

徐成輔 부산교육대학교 과학교육연구소 1985 科學敎育硏究 Vol.10 No.-

먼저 점p에서 곡면의 S의 normal vector field, 즉 p에서 tangent plane Tp(s)에 수직인 직선을 소개하고, 2次形式 I,II, 그리고 III즉, 점X(u,v)에서 SD의 제1, 제2, 제3 기본형식을 정의한다 이래서 제 1기본형식 I과 제2기본 형식 II사이의 기하학적인 해석 즉 κcos0=II/I를 갖는다 다음 T의 方向에 있는 곡면 S의 normal curvature를 κα(0)cos0=ㅣldu2+2mdudv+ndu2/Edu2_2Fdudv+Gdv2이고, 곡면 S⊂E3위에서 여라기지 方向에 있는 normal courvature는 거기에 대응하는 normal sections가 어떻게 보이는가를 그림으로 그려서 合利的인 評價를 하도록 한다. 더우기 l=|Xu'Xu,Xuu|/√EG-F2,m=|Xu,Xv, Xuv|/√FG-F2and n=|Xu,Xu,Xuv|√EG-F2를 사용해서, In-m>0, In-m<0 또는 In-m=0에 따라 점 X(u,v)에서 곡면 S의 楕圓點 雙曲點, 또는 抛物點을 소개할까 한다. 制輛數 가진 몇개 변수의 함수에 대한 극치문제를 연구하기 위한 Lagrange mutipliers의 방법을 이미 알고 있다. 마지막으로 다음과 같은 세개의 중요한 사실을 얻을 수 있다. (1) 타우너점을 갖는 곡면의 principa curvatures는 같은 보호를 갖는다. (2) 쌍곡점을 갖는 곡면의 principa curvatures는 서로 반대부호를 갖는다. (3) 포물선을 갖는 곡면의 principa curvatures는 오직 하나가 zero이다. En premier lieu, nous introdisions le champ du vecteur normal d'une surface S a' un point p, c'est-a'-dire, la ligne orthogonale vers le plan tangent Tp(S) at p, etde'ginissons les formes quadratiques I,II et III, autrement dit, la deuxie`me et la troisie`me forme fordmentale, de S a`point X(u,v). Ainsi, nous avons I'inter-pte`tation ge'me'trique enter la prem premile're forme fondamentale I et la deuxie'me forme fondametale II; κcos Θ=I/II. En suitem nous de'finissons la coubure normal K de la surface S en direction T par κα(0) cos 0=K ou' Kα(0) ocs0= ldu2+2mdudp+ndu2/Edu2+2Fdudv+Gdv2 et faisons l'estmation reisonnable de la courbure normale dans les diverses directions sur une surface S⊂E3 par peinturant ce gue les sections normales correspondantes regarent comme. D'ailleurs, se servant de l=|xu,Xv,Xuu|/√EG-F2, m=|Xu,Xu,Xuv|/√EG-F2 et n= |Xu,Xu,Xuv|/√EG-F2 Nous introduisons le point elliptique, hyperbolique, ou parabolique de la surface S a'le point X(u,v), selon que in-m>0, in-m=0. Nous avous de'jara`su la me'thode des Lagrange multiplieurs afin d'e`tudier les proble'ems extre'mes pour les fonctions des plusierurs variables avec une constrainte. Ainsi, nous sommes maintenant en position de trouver les valures des deux extre'mes de la courbure normale Kʼn par s;appliquent la meㄱ3thode des lagrange multiplieurs, et nous amenons les courbures principales des deux extre`mes de la courbure normale K. Nous pouvons enfin abtenir les trois faits importants suivants; (1) Les coubures principales d'une surface dont les points sont elliptique, ont le me'e signe. (2) Les coubures principales d'une surface dont les points sont parabolique, est ze'ro.

제1회 한새벌 초등학교 수학경시대회 결과 분석

서성보 부산교육대학교 과학교육연구소 1998 科學敎育硏究 Vol.23 No.-

This Mathematical contest was held at four classrooms in Pusan National University of Education, November 15, 1997. Among 256 Elemetary schools in Pusan, 198 children with a talent in mathematics, in the sixth grad (133 boys and 65 girls) of 126 schools, participated in this mathematics contest. A number of questions in five small domains and distributions of their result are as follows: two questions on divisors the greatest common divisor and the least common multiple have 39 marks, one question on the the four arithmetical operations has 21 marks, two questions on the solar calendar have 32 marks, three questions on the magic square have 36 marks, and three questions on the areas and properties of figures have 72 marks. Thus the above 11 questions in five small domains have a total of 200 marks. Total average of all questions that the 133 boys examined was 142.57 marks (71.29%), and the total average of all questions that the 65 girls examined was 143.12 marks (71.56%). And the questions in five small domains had the following marks in order; 'One questions on divisors the greatest common divisor and the least common multiple" had marks of 81.32%, "Three questions on the magic square" had marks of 79.76%, "Three questions on the areas and properties of figures" had marks of 58.73%, and finally "Two questions on the solar calendar" had marks of 58.63%.

국민학교 기하도형의 주요 학습요소 해설 : 4, 5, 6학년 수학교과서를 중심으로

서성보 부산교육대학교 초등교육연구소 1993 초등교육연구 Vol.3 No.-

인지이론을 근거로 한 수학학습 방법 탐색과 지도의 실제

서성보,박성택 부산교육대학교 초등교육연구소 1996 초등교육연구 Vol.9 No.-

The purpose of this to sequentialize Mathematics-learning contents, and to explore teaching-learning model for mathematics, which on the basis of the theory of cognitive development and the period of conservation for children. The results of subjects are as follows: (1) Cognitive development can be achieved by constant space and Mathematics knowledge is obtained by the interaction of experience and reason. (2) The stages of cognitive development for children to teach mathematics systematically and orderly. (3) The learning effect of mathematical concepts occurs when this coincides with the period of conservation formation for children. (4) Mathematics Curriculum of Elementary School in Korea matches with the experimental research about the period of Piaget's conservation formation. (5) Mathematics learning is to be centered on learning by experience such as observation, operation, experiment and actual measurement.

The Lie Differentiation of Weyl's Conformal Curvature Tensor by the Conformal Killing Vector

徐成輔 부산교육대학교 과학교육연구소 1986 科學敎育硏究 Vol.11 No.-

We first define the Riemannian curvature tensor R_(λμυ)^κ of (1, 3) type and the conformal correspondence g=e^(2β)g of two Riemannian metrics g and g, and from these facts we reduce the Weyl's conformal curvature tensor C_(λμυ)^κ of (1, 3) type. Thus we prove that the Lie differentiation of C_(λμυ)^κ by the conformal killing vector field X is zero.

아르키메데스의 求積에 관하여

徐成輔 부산교육대학교 과학교육연구소 1990 科學敎育硏究 Vol.15 No.-

The first problems occurring in the History of the calculus were concerned with the computation of areas, volumes, and lengths of arcs, and in their treatment one finds evidence of the two assumptions about the divisibility of magnitudes that we considered. One of the earliest important contributions to the problem of squaring the circle was that of Antiphon the Sophist(ca. 430 B. C.). He advanced the idea that by successively doubling the number of sides of a regular polygon inscribed in a circle, the difference in area between the circle and the polygon would at last be exhausted. The method exhausition is usually credited to Eudoxus(ca. 370 B. C.), and assumes the infinite divisibility of magnitudes, and has, as a basis, the proposition : If from any magnitude there be subtracted a part not less than its half, from the remainder another part not less than its half, and so on, there will at length remain a magnitude less than any preassgined magnitude of the same kind. On the other hand, in Physics, the equilibrium is said to be a state of rest or balance due to the equal action of opposing forces. The method of equilibrium used its theory is the Archimedes' way of discovering the formula for the area of a parabolic segment and the volume of a sphere. Thus using particular property of the parabola, Archimedis(287∼212 B. C) had found the area of the parabolric segment by the two methods of exhaution and equilibruim ; the area of a parabolic segment is four-thirds that of the inscribed triangle having the same base and having its opposite vertex at the point where the tangent is parallel to the base. Moreover, Archimedes had established the fact that the volume of a sphere is 2/3 that of the circumscribed cylinder by using the method of equilibium. Like this, the works of Archimedes are masterpieces of mathematical exposition and to a remarkable extent resemble modern journal articles.

E^4 空間에 있는 曲面들의 G-全曲率

徐成輔,姜龍洙 부산교육대학교 과학교육연구소 1987 科學敎育硏究 Vol.12 No.-

우리는 보통 E^3空間內에 잇는 어떤 曲面에 대한 Gauss 曲率 K와 平均曲率 α를 名名 K=∑det(A^r_(ij))와 α=√<H,H>(단, H=1/n ∑A^r_(ij)e_r)로 定義하고 있는데, B.Y.Chen은 單位法束 Bν 위의 한 點(p,e)에서 潛入寫像 X:M^n→E^m의 i번째 平均曲率 K_i(p,e)를 使用하여 i번째 G-金曲率 G_i(x,p,g,k)와 T_i(x,g,k), 그리고 i번째 平均曲率 G-絶對全曲率 K^*_i(x,p,g,k)와 TA_i(x,g,k)를 定義하고, 더 나아가서 T_i(x,g,k)와 T_(i+1)(x,g,k) 사이의 關係式까지 硏究하였다. 더우기 S.S.Chern과 R.K.Lashof는 潛入寫像 X:M^n→E^m에 대하여 TA_n(x,1,1)<3C_(m-1)이면 M^n는 하나의 n-球와 位相同形임을 證明하였다. 本 論文에서는 우선 M^2위에 있는 點P에서 하나의 Frenet frame을 定義하고 이것을 바탕으로 하여 K_1(p,e)와 K_2(p,e)을 誘導한 다음 Bν위에서 G-金曲率과 絶對全曲率에 관한 여러가지 性質을 調査 硏究하였다. 第3章에서는, E^4空間에서 det(A^3_(ij))≥det(A^4_(ij))(i,j=1,2)이고 A^3_11A^4_22+A^4_11A^3_22-2A^3_12A^4_21=0인 Frenet frame e_1, e_2, e_3, e_4를 擇하면 K_1(p,e)=1/2[(a^3_11+A^3_22)cosθ+(A^4_11+A^4_22)sinθ]이고 K_2(p,e)=λ(p)cos^2θ+μ(p)sin^2θ인것을 증명하였으며, 이것을 利用해서 T_1(x,1,2)=πㄷ∫_Mα^2dν와 T_2(x,1,1)=2π^2χ(M)等과 같은 關係式을 얻었다.

The Even Dimensional Case in the Gauss-Bonnet Formula : La cas de Dimension Pair dans la Gauss-Bonnet Formule

徐成輔 부산교육대학교 과학교육연구소 1986 科學敎育硏究 Vol.11 No.-

D´abord, soit ω une forme du degre´ n-1 dans une varie´te´ M, et D un domaine orie nte´ compact dans M borns´ par une courbe doucereuse sectionnelle γ. Alors nous pouvons gagner le the´ore´me de Stokes ?? dω = ??dω. D´ailleurs, de cet the´ore´me, nous obtenons la formule de Gauss-Bonnet ?? kgds + ?? KdA + ?? (π-αi) = 2πx, ou´ kg est la courbure ge´odesigue de la courbe γ, π-αi sont les angles exte´rieures aux sommets de γ, et x est la caracte´ristique d´ Euler du domaine D. Si la courbe γ n´ a Pas de sommet, la formule se simplifie a´ ?? kgds + ??KdA = 2πx. Particulie´rement, si D est la surface entie´re M, nous avons ??KdA = 2πx. Maintenant, conside´rant la forrmeγ de´finie par 4πΩ = -∈_(i1i2)Ω_(i1i2), nous gagnons 2πΩ = -K_(ω1)A_(ω2). De plus, si nous construisons deux formes diffe´rentielles Φ_0 du degre´ 1 et Ψ_0 du degre´ 2 : Φ_0 = ∈_(i1i2)u_(i1)u_(i2) et Ψ_0 = ∈_(i1i2)Ω_(i1i2), alors nous avons dΦ_0 = 1/2 Ψ_0. Ainsi nous Prouvons ??-Ω = x(M), c´est-a´-dire 1/2π??KdA = x(M). D´un autre co^te´, soit M une varie´te´ Riemannienne oriente´e du dimension pair, n = 2p. Et quand nous conside´rons la forme Ω de´finie Par Ω = (-1)^p 1/2^(2p)π^pP!∈_(i1i2)Ω_(i1i2)Ω_(i3i4)Ω_(i2p-1i2p), nous pouvons e´crire la formule de Gauss-Bonnet par forme de ?? Ω = x. Afin d´acque´rir cette formule, nous construisons deux ensembles des formes differentielles suivantes : Φk = ∈i_1…i_p u_i Θi_2…Θi_(2p)-_2k Ωi_2p-_2k+1 i_2p-_2k+2…Ωi_2p-i_2p, Ψk = ∈i_1…i_2p Ωi_1i_2 Θi_3…Θi_(2p)-_2k Ωi_2p-_2k+1 i_2p-_2k+2…Ωi_2p-1i_2p, ou´ k = 0,1,2,…, p-1, et les formes Φk sont de degre´ sont de degre´ 2p-1 et Ψk sont de degre´ 2??. De deux formes dessus, nous pouvons de´river la relation re´currente suivante : dΦk = Ψ_(k-1)+2p-2k+1/2(k+1)Ψ_k. En particulier, ils´ensuit que Ω est la de´rivative e´xte´rieure d´une forme Ⅱ: Ω = (-1)^p 1/2^(2p)π^pP!Ψ_(p-1) = Ⅱ, ou´ Ⅱ = 1/π^p??(-1)^(m+1)1/1·3…(2p-2m-1)m!2^(p+m)Φ_m Mais, a´ pre´sent, le champ vectoriel de´finidans M^n, une sou-varie´te´ V^n, avec la borne X^2 ou´ Z est le (n-1)-cycle dimensionel forme´ par toutes unite´s vecteurs au travers du point 0. En conse´quence, 1´inte´gral de Ω au dessus de M^n est e´gal au inte´gral au-dessus de V^n. Usant le the´ore`me de Stokes, nous obtenons ??Ω = ?? Ω = X??π = -X1/1·3…(2p-1)2^p·π^p??Φ_0. Ainsi, de la de´finition of Φ_0 nous avons Φ_0 = (2p-1)!?/(-1)^iθ_1…θ_-1u_iθ_(i+1)…θ2p. Si nous de´notons dV en le volume e´le´ment du (2p-1)-dimensionel unite´ sphe´re, nous gagnons dV = ??(-1)^iθ_1…θ_(i-1)u_iθ_(i+1)… θ_2p, duquel nous de´duisons Φ_0 = (2p-1)! dV. Donc nous obtenous la formule ge´ne´ralise´e de Gauss-Bonnet ??Ω = -x.

Ca test를 이용해 PEN 기판위에 증착된 (SiO₂)₃₀(ZnO)₇₀ 박막의 수분방지 특성

서성보,김해진,손선영,김화민 한국물리학회 2012 새물리 Vol.62 No.3

In this study, high density and low temperature inorganic thin composite SZO films consisting of (SiO_2)_(30)(ZnO)_(70) were deposited as barrier layers by using a facing target sputtering system, and their WVTRs were quantitatively caculated by using the Ca test. PEN with a SZO (FTS) film showed an excellent barrier property of 3.34 × 10^(-3) g/㎡-day, In addition, PEN with deposited a 400 nm SZO(FTS) film showed a superior barrier property of 2.45 × 10^(-4) g/㎡-day. In particular, the value of the WVTR measured by using the Ca test, below 5 ×10^(-3)g/㎡-day, was suitable for its use as a barrier film. The results were shown to have high reliability based on an error factor of about 7.5 \% between the MOCON test and the Ca test. 본 연구에서는 박막의 고밀도와 저온증착이 가능한 대향 타겟식스퍼터링(Facing Target sputtering, FTS) 방법을 사용하여 SiO_2가 30wt.% 도핑된 ZnO(SZO) 혼합 무기 박막을 제작하였으며, Ca test를 통해박막의 수분투과율(Water Vapor Transmission rate, WVTR)을 측정하였다. PEN 기판위에 증착된 SZO 무기박막은 Ca test를 통해 3.34 ×10^(-3) g/㎡-day의 낮은 수분투과율을 나타내었다. 또한 PEN 기판양쪽에 SZO를 증착할 경우 한쪽 면에 증착했을 때 보다 더욱 낮은 2.45 × 10^(-4) g/㎡-day의 결과를 보여준다. Ca test는 MOCON test의 한계측정치인 5 × 10^(-3) g/㎡-day 이하의수분투과율 측정이 가능하며, 실제 MOCON test와 수분투과율 비교시오차율은 약 7.5 %의 신뢰성도 높은 것으로 나타났다.