Difference between revisions of "Weingarten derivational formulas"
From Encyclopedia of Mathematics
(Importing text file) |
Ulf Rehmann (talk | contribs) m (tex encoded by computer) |
||
| Line 1: | Line 1: | ||
| − | + | <!-- | |
| + | w0976201.png | ||
| + | $#A+1 = 10 n = 0 | ||
| + | $#C+1 = 10 : ~/encyclopedia/old_files/data/W097/W.0907620 Weingarten derivational formulas | ||
| + | Automatically converted into TeX, above some diagnostics. | ||
| + | Please remove this comment and the {{TEX|auto}} line below, | ||
| + | if TeX found to be correct. | ||
| + | --> | ||
| − | + | {{TEX|auto}} | |
| + | {{TEX|done}} | ||
| − | + | Formulas yielding the expansion of the derivative of the unit normal vector to a surface in terms of the first derivatives of the position vector of this surface. Let $ \mathbf r = {\mathbf r } ( u, v) $ | |
| + | be the position vector of the surface, let $ \mathbf n $ | ||
| + | be the unit normal vector and let $ E $, | ||
| + | $ F $, | ||
| + | $ G $, | ||
| + | $ L $, | ||
| + | $ M $, | ||
| + | $ N $ | ||
| + | be the coefficients of the first and second fundamental forms of the surface, respectively; the Weingarten derivational formulas will then take the form | ||
| + | |||
| + | $$ | ||
| + | \mathbf n _ {u} = | ||
| + | \frac{FM- GL }{EG - F ^ { 2 } } | ||
| + | |||
| + | \mathbf r _ {u} + FL- | ||
| + | \frac{EM}{EG- F ^ { 2 } } | ||
| + | \mathbf r _ {v} , | ||
| + | $$ | ||
| + | |||
| + | $$ | ||
| + | \mathbf n _ {v} = FN- | ||
| + | \frac{GM}{EG- F ^ { 2 } } | ||
| + | \mathbf r _ {u} + FM- | ||
| + | \frac{EN}{EG- F ^ { 2 } } | ||
| + | \mathbf r _ {v} . | ||
| + | $$ | ||
The formulas were established in 1861 by J. Weingarten. | The formulas were established in 1861 by J. Weingarten. | ||
| Line 9: | Line 42: | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> P.K. Rashevskii, "A course of differential geometry" , Moscow (1956) (In Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> P.K. Rashevskii, "A course of differential geometry" , Moscow (1956) (In Russian)</TD></TR></table> | ||
| − | |||
| − | |||
====Comments==== | ====Comments==== | ||
| − | |||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> N.J. Hicks, "Notes on differential geometry" , v. Nostrand (1965)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> N.J. Hicks, "Notes on differential geometry" , v. Nostrand (1965)</TD></TR></table> | ||
Latest revision as of 08:29, 6 June 2020
Formulas yielding the expansion of the derivative of the unit normal vector to a surface in terms of the first derivatives of the position vector of this surface. Let $ \mathbf r = {\mathbf r } ( u, v) $
be the position vector of the surface, let $ \mathbf n $
be the unit normal vector and let $ E $,
$ F $,
$ G $,
$ L $,
$ M $,
$ N $
be the coefficients of the first and second fundamental forms of the surface, respectively; the Weingarten derivational formulas will then take the form
$$ \mathbf n _ {u} = \frac{FM- GL }{EG - F ^ { 2 } } \mathbf r _ {u} + FL- \frac{EM}{EG- F ^ { 2 } } \mathbf r _ {v} , $$
$$ \mathbf n _ {v} = FN- \frac{GM}{EG- F ^ { 2 } } \mathbf r _ {u} + FM- \frac{EN}{EG- F ^ { 2 } } \mathbf r _ {v} . $$
The formulas were established in 1861 by J. Weingarten.
References
| [1] | P.K. Rashevskii, "A course of differential geometry" , Moscow (1956) (In Russian) |
Comments
References
| [a1] | W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973) |
| [a2] | N.J. Hicks, "Notes on differential geometry" , v. Nostrand (1965) |
How to Cite This Entry:
Weingarten derivational formulas. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weingarten_derivational_formulas&oldid=49201
Weingarten derivational formulas. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weingarten_derivational_formulas&oldid=49201
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article