"라플라스-벨트라미 연산자"의 두 판 사이의 차이

2012년 1월 15일 (일) 09:33 판

리만다양체의 메트릭이 \(g_{ij}\)로 주어지는 경우
\((g^{ij})=(g_{ij})^{-1}\)
라플라시안
\(\Delta f=\nabla_i \nabla^i f =\frac{1}{\sqrt{\det g}} \frac{\partial }{\partial x^j}\left(g^{jk}\sqrt{\det g}\frac{\partial f}{\partial x^k}\right) = g^{jk}\frac{\partial^2 f}{\partial x^j \partial x^k} + \frac{\partial g^{jk}}{\partial x^j} \frac{\partial f}{\partial x^k} + \frac12 g^{jk}g^{il}\frac{\partial g_{il}}{\partial x^j}\frac{\partial f}{\partial x^k}\)
곡면의 경우 \(E=g_{11}\), \(F=g_{12}=g_{21}\), \(G=g_{22}\)

\(F=0\)인 경우
\(\Delta f=\frac{1}{\sqrt{EG}}\left( \frac{\partial }{\partial x^1}\left(\sqrt{\frac{G}{E}}\frac{\partial f}{\partial x^1}\right)+\frac{\partial }{\partial x^2}\left(\sqrt{\frac{E}{G}}\frac{\partial f}{\partial x^2}\right)\right)\)

\(\Delta f = \frac{\partial^2f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2}\)

극좌표계
\(E=1\), \(G=0\), \(F=r^2\)
\(\sqrt{EG}=r\)
\(\Delta f = {1 \over r} {\partial \over \partial r} \left( r {\partial f \over \partial r} \right) + {1 \over r^2} {\partial^2 f \over \partial \theta^2}={1 \over r} {\partial f \over \partial r}+ {\partial^2 f \over \partial r^2}+{1 \over r^2} {\partial^2 f \over \partial \theta^2}\)

구면(sphere)
\(\Delta f ={1 \over r^2 }({\partial^2 f \over \partial \theta^2} +\cot\theta {\partial f \over \partial \theta} + \frac{1}{ \sin^2 \theta}{\partial^2 f \over \partial \phi^2})\)

구면좌표계
\(\Delta f = {1 \over r^2} {\partial \over \partial r} \left( r^2 {\partial f \over \partial r} \right) + {1 \over r^2 \sin \theta} {\partial \over \partial \theta} \left( \sin \theta {\partial f \over \partial \theta} \right) + {1 \over r^2 \sin^2 \theta} {\partial^2 f \over \partial \varphi^2}\)

@@ 90번째 줄: / 90번째 줄: @@
 * [[편미분방정식]]
+<h5>매스매티카 파일 및 계산 리소스</h5>
+*
+* 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]
@@ 112번째 줄: / 129번째 줄: @@
 * http://en.wikipedia.org/wiki/List_of_formulas_in_Riemannian_geometry
 * http://en.wikipedia.org/wiki/Laplace_operators_in_differential_geometry
-* http://mathworld.wolfram.com/Laplacian.html[http://www.wolframalpha.com/input/?i=laplacian ]
+* http://mathworld.wolfram.com/Laplacian.html
 * [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]<br>
 ** http://dlmf.nist.gov/1.5