"회전으로 얻어지는 곡면"의 두 판 사이의 차이

2021년 2월 17일 (수) 03:33 기준 최신판

개요

평면 상의 곡선이 \((f(v), g(v))\) 로 매개화될 때, x축 또는 y축을 기준으로 회전하여 얻어지는 곡면
3차원상에 놓여 있는 매개화된 곡면을 얻는다
y축에 대하여 회전하는 경우, 매개화는 \(\mathbf{x}(u,v)=(f(v) \cos (u),f(v) \sin (u),g(v))\) 로 주어진다

예

곡선 를 y축에 대하여 회전하여 곡면 를 얻는다

제1기본형식

곡면의 매개화가 \(\mathbf{x}(u,v)=(f(v) \cos (u),f(v) \sin (u),g(v))\) 로 주어졌다고 하자
\(E=f(v)^2\)
\(F=0\)
\(G=f'(v)^2+g'(v)^2\)

크리스토펠 기호

\[\begin{array}{ll} \Gamma _{11}^1 & 0 \\ \Gamma _{12}^1 & \frac{f'(v)}{f(v)} \\ \Gamma _{21}^1 & \frac{f'(v)}{f(v)} \\ \Gamma _{22}^1 & 0 \\ \Gamma _{11}^2 & -\frac{f(v) f'(v)}{f'(v)^2+g'(v)^2} \\ \Gamma _{12}^2 & 0 \\ \Gamma _{21}^2 & 0 \\ \Gamma _{22}^2 & \frac{f'(v) f''(v)+g'(v) g''(v)}{f'(v)^2+g'(v)^2} \end{array}\]

리만 곡률 텐서

\[\begin{array}{ll} \begin{array}{ll} R_{111}^1 & 0 \\ R_{112}^1 & 0 \end{array} & \begin{array}{ll} R_{121}^1 & 0 \\ R_{122}^1 & 0 \end{array} \\ \begin{array}{ll} R_{211}^1 & 0 \\ R_{212}^1 & \frac{g'(v) \left(f'(v) g''(v)-f''(v) g'(v)\right)}{f(v) \left(f'(v)^2+g'(v)^2\right)} \end{array} & \begin{array}{ll} R_{221}^1 & \frac{g'(v) \left(f''(v) g'(v)-f'(v) g''(v)\right)}{f(v) \left(f'(v)^2+g'(v)^2\right)} \\ R_{222}^1 & 0 \end{array} \\ \begin{array}{ll} R_{111}^2 & 0 \\ R_{112}^2 & \frac{f(v) g'(v) \left(f''(v) g'(v)-f'(v) g''(v)\right)}{\left(f'(v)^2+g'(v)^2\right)^2} \end{array} & \begin{array}{ll} R_{121}^2 & \frac{f(v) g'(v) \left(f'(v) g''(v)-f''(v) g'(v)\right)}{\left(f'(v)^2+g'(v)^2\right)^2} \\ R_{122}^2 & 0 \end{array} \\ \begin{array}{ll} R_{211}^2 & 0 \\ R_{212}^2 & 0 \end{array} & \begin{array}{ll} R_{221}^2 & 0 \\ R_{222}^2 & 0 \end{array} \end{array}\]

가우스 곡률

\[K=\frac{g'(v) \left(f'(v) g''(v)-f''(v) g'(v)\right)}{f(v) \left(f'(v)^2+g'(v)^2\right)^2}\]

역사

메모

매스매티카 파일 및 계산 리소스

https://docs.google.com/leaf?id=0B8XXo8Tve1cxZWM2ZDQzYzktMjhmMi00ZmVhLTg5N2MtZjlhYTg5OWQzNzdi&sort=name&layout=list&num=50

사전 형태의 자료

메타데이터

위키데이터

ID : Q849425

Spacy 패턴 목록

[{'LOWER': 'surface'}, {'LOWER': 'of'}, {'LEMMA': 'revolution'}]

@@ 1번째 줄: / 1번째 줄: @@
-==이 항목의 스프링노트 원문주소==
-* [[회전으로 얻어지는 곡면]]
 ==개요==
@@ 13번째 줄: / 5번째 줄: @@
 * y축에 대하여 회전하는 경우, 매개화는 <math>\mathbf{x}(u,v)=(f(v) \cos (u),f(v) \sin (u),g(v))</math> 로 주어진다
 ==예==
+곡선
+[[파일:회전으로 얻어지는 곡면1.png]]
+를 y축에 대하여 회전하여 곡면
+[[파일:회전으로 얻어지는 곡면2.png]]
+를 얻는다
-[https://lh6.googleusercontent.com/FgjMek6RDQC0frijOLlw8_ppy_unBujvM1rw6P4NTheNNpT1Quxz2qbUcApZRkAOrwjlUAh-ffXFc4dMoux0n57kSAepwcWArJs=w1600 ]
-[https://lh6.googleusercontent.com/Ax6kfcRTbsIIBmc8jaVVVJbDUHL6PE3DpUfaEyh2ceruUuEnX_9Hw9ALVRKF8WT4DM5dw8Icql_dRywhAy5zrml-tRsBw5Vz6I4=w1600 ]를 y축에 대하여 회전하여 곡면[https://lh6.googleusercontent.com/lHIePZ5whm_y5Wa_g1Op4h5Uf8O_S7QTFtMOTOFQlcx0lEEGN7yYmeJ03_BO7_V9w-4bLDmJ2BivvrrSCwvwlXuI7IgFo_ZrgHg=w1600 ][https://lh5.googleusercontent.com/i9aC5ZG2IXeRcgkUfIJBJv4VVA70CTHcKLRNtlJ_8rtwNHWC-Sk_bkGXqcHAp1EGdfasgOpV82Pc1sKfGuZKtUfSrN3Ew7Cqwg4=w1600 ]를 얻는다
 ==제1기본형식==
@@ 34번째 줄: / 27번째 줄: @@
 * <math>G=f'(v)^2+g'(v)^2</math>
 ==크리스토펠 기호==
+:<math>\begin{array}{ll}  \Gamma _{11}^1 & 0 \\  \Gamma _{12}^1 & \frac{f'(v)}{f(v)} \\  \Gamma _{21}^1 & \frac{f'(v)}{f(v)} \\  \Gamma _{22}^1 & 0 \\  \Gamma _{11}^2 & -\frac{f(v) f'(v)}{f'(v)^2+g'(v)^2} \\  \Gamma _{12}^2 & 0 \\  \Gamma _{21}^2 & 0 \\  \Gamma _{22}^2 & \frac{f'(v) f''(v)+g'(v) g''(v)}{f'(v)^2+g'(v)^2} \end{array}</math>
-<math>\begin{array}{ll}  \Gamma _{11}^1 & 0 \\  \Gamma _{12}^1 & \frac{f'(v)}{f(v)} \\  \Gamma _{21}^1 & \frac{f'(v)}{f(v)} \\  \Gamma _{22}^1 & 0 \\  \Gamma _{11}^2 & -\frac{f(v) f'(v)}{f'(v)^2+g'(v)^2} \\  \Gamma _{12}^2 & 0 \\  \Gamma _{21}^2 & 0 \\  \Gamma _{22}^2 & \frac{f'(v) f''(v)+g'(v) g''(v)}{f'(v)^2+g'(v)^2} \end{array}</math>
 ==리만 곡률 텐서==
+:<math>\begin{array}{ll}   \begin{array}{ll}  R_{111}^1 & 0 \\  R_{112}^1 & 0 \end{array}  &  \begin{array}{ll}  R_{121}^1 & 0 \\  R_{122}^1 & 0 \end{array}  \\   \begin{array}{ll}  R_{211}^1 & 0 \\  R_{212}^1 & \frac{g'(v) \left(f'(v) g''(v)-f''(v) g'(v)\right)}{f(v) \left(f'(v)^2+g'(v)^2\right)} \end{array}  &  \begin{array}{ll}  R_{221}^1 & \frac{g'(v) \left(f''(v) g'(v)-f'(v) g''(v)\right)}{f(v) \left(f'(v)^2+g'(v)^2\right)} \\  R_{222}^1 & 0 \end{array}  \\   \begin{array}{ll}  R_{111}^2 & 0 \\  R_{112}^2 & \frac{f(v) g'(v) \left(f''(v) g'(v)-f'(v) g''(v)\right)}{\left(f'(v)^2+g'(v)^2\right)^2} \end{array}  &  \begin{array}{ll}  R_{121}^2 & \frac{f(v) g'(v) \left(f'(v) g''(v)-f''(v) g'(v)\right)}{\left(f'(v)^2+g'(v)^2\right)^2} \\  R_{122}^2 & 0 \end{array}  \\   \begin{array}{ll}  R_{211}^2 & 0 \\  R_{212}^2 & 0 \end{array}  &  \begin{array}{ll}  R_{221}^2 & 0 \\  R_{222}^2 & 0 \end{array}  \end{array}</math>
-<math>\begin{array}{ll}   \begin{array}{ll}  R_{111}^1 & 0 \\  R_{112}^1 & 0 \end{array}  &  \begin{array}{ll}  R_{121}^1 & 0 \\  R_{122}^1 & 0 \end{array}  \\   \begin{array}{ll}  R_{211}^1 & 0 \\  R_{212}^1 & \frac{g'(v) \left(f'(v) g''(v)-f''(v) g'(v)\right)}{f(v) \left(f'(v)^2+g'(v)^2\right)} \end{array}  &  \begin{array}{ll}  R_{221}^1 & \frac{g'(v) \left(f''(v) g'(v)-f'(v) g''(v)\right)}{f(v) \left(f'(v)^2+g'(v)^2\right)} \\  R_{222}^1 & 0 \end{array}  \\   \begin{array}{ll}  R_{111}^2 & 0 \\  R_{112}^2 & \frac{f(v) g'(v) \left(f''(v) g'(v)-f'(v) g''(v)\right)}{\left(f'(v)^2+g'(v)^2\right)^2} \end{array}  &  \begin{array}{ll}  R_{121}^2 & \frac{f(v) g'(v) \left(f'(v) g''(v)-f''(v) g'(v)\right)}{\left(f'(v)^2+g'(v)^2\right)^2} \\  R_{122}^2 & 0 \end{array}  \\   \begin{array}{ll}  R_{211}^2 & 0 \\  R_{212}^2 & 0 \end{array}  &  \begin{array}{ll}  R_{221}^2 & 0 \\  R_{222}^2 & 0 \end{array}  \end{array}</math>
 ==가우스 곡률==
+:<math>K=\frac{g'(v) \left(f'(v) g''(v)-f''(v) g'(v)\right)}{f(v) \left(f'(v)^2+g'(v)^2\right)^2}</math>
-<math>K=\frac{g'(v) \left(f'(v) g''(v)-f''(v) g'(v)\right)}{f(v) \left(f'(v)^2+g'(v)^2\right)^2}</math>
 ==역사==
 * http://www.google.com/search?hl=en&tbs=tl:1&q=
 * [http://jeff560.tripod.com/mathword.html Earliest Known Uses of Some of the Words of Mathematics]
 * [http://jeff560.tripod.com/mathsym.html Earliest Uses of Various Mathematical Symbols]
-* [[수학사연표 (역사)|수학사연표]]
+* [[수학사 연표]]
 ==메모==
 ==관련된 항목들==
 ==매스매티카 파일 및 계산 리소스==
 * https://docs.google.com/leaf?id=0B8XXo8Tve1cxZWM2ZDQzYzktMjhmMi00ZmVhLTg5N2MtZjlhYTg5OWQzNzdi&sort=name&layout=list&num=50
-* http://www.wolframalpha.com/input/?i=
-* http://functions.wolfram.com/
-* [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]
-* [http://people.math.sfu.ca/%7Ecbm/aands/toc.htm Abramowitz and Stegun Handbook of mathematical functions]
-* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
-* [http://numbers.computation.free.fr/Constants/constants.html Numbers, constants and computation]
-* [https://docs.google.com/open?id=0B8XXo8Tve1cxMWI0NzNjYWUtNmIwZi00YzhkLTkzNzQtMDMwYmVmYmIxNmIw 매스매티카 파일 목록]
-==수학용어번역==
-* 단어사전 http://www.google.com/dictionary?langpair=en|ko&q=
+==사전 형태의 자료==
-* 발음사전 http://www.forvo.com/search/
-* [http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr= 대한수학회 수학 학술 용어집]<br>
-** http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
-* [http://www.nktech.net/science/term/term_l.jsp?l_mode=cate&s_code_cd=MA 남·북한수학용어비교]
-* [http://kms.or.kr/home/kor/board/bulletin_list_subject.asp?bulletinid=%7BD6048897-56F9-43D7-8BB6-50B362D1243A%7D&boardname=%BC%F6%C7%D0%BF%EB%BE%EE%C5%E4%B7%D0%B9%E6&globalmenu=7&localmenu=4 대한수학회 수학용어한글화 게시판]
-==사전 형태의 자료==
-* http://ko.wikipedia.org/wiki/
 * http://en.wikipedia.org/wiki/Surface_of_revolution
 * http://www.wolframalpha.com/input/?i=surface+of+revolution
 [[분류:미분기하학]]
+[[분류:곡면]]
+==메타데이터==
+===위키데이터===
+* ID :  [https://www.wikidata.org/wiki/Q849425 Q849425]
+===Spacy 패턴 목록===
+* [{'LOWER': 'surface'}, {'LOWER': 'of'}, {'LEMMA': 'revolution'}]