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Cauchy's theorem on polyhedra: Two closed convex polyhedra are congruent if their true faces, edges and vertices can be put in an incidence-preserving one-to-one correspondence in such a way that corresponding faces are congruent. This is the first theorem about the unique definition of convex surfaces, since the polyhedra of which it speaks are isometric in the sense of an intrinsic metric. The Cauchy theorem is a special case of the theorem stating that every closed convex surface is uniquely defined by its metric (see [[#References|[4]]]).
 
Cauchy's theorem on polyhedra: Two closed convex polyhedra are congruent if their true faces, edges and vertices can be put in an incidence-preserving one-to-one correspondence in such a way that corresponding faces are congruent. This is the first theorem about the unique definition of convex surfaces, since the polyhedra of which it speaks are isometric in the sense of an intrinsic metric. The Cauchy theorem is a special case of the theorem stating that every closed convex surface is uniquely defined by its metric (see [[#References|[4]]]).
  
Line 8: Line 20:
 
''E.V. Shikin''
 
''E.V. Shikin''
  
Cauchy's intermediate-value theorem for continuous functions on closed intervals: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c0209901.png" /> be a continuous real-valued function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c0209902.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c0209903.png" /> be a number between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c0209904.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c0209905.png" />. Then there is a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c0209906.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c0209907.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c0209908.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c0209909.png" /> have different signs, then there is a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c02099010.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c02099011.png" />. This version of Cauchy's theorem is used to determine intervals in which a function necessarily has zeros. It follows from Cauchy's theorem that the image of an interval on the real line under a continuous mapping into the real line is also an interval. The theorem can be generalized to topological spaces: Any continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c02099012.png" /> defined on a connected topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c02099013.png" /> and assuming two distinct values, also assumes any value between them; hence the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c02099014.png" /> is also an interval on the real line.
+
Cauchy's intermediate-value theorem for continuous functions on closed intervals: Let $  f $
 +
be a continuous real-valued function on $  [a, b] $
 +
and let $  C $
 +
be a number between $  f (a) $
 +
and $  f (b) $.  
 +
Then there is a point $  \xi \in [a, b] $
 +
such that $  f ( \xi ) = C $.  
 +
In particular, if $  f (a) $
 +
and $  f (b) $
 +
have different signs, then there is a point $  \xi $
 +
such that $  f ( \xi ) = 0 $.  
 +
This version of Cauchy's theorem is used to determine intervals in which a function necessarily has zeros. It follows from Cauchy's theorem that the image of an interval on the real line under a continuous mapping into the real line is also an interval. The theorem can be generalized to topological spaces: Any continuous function $  f: X \rightarrow \mathbf R  ^ {1} $
 +
defined on a connected topological space $  X $
 +
and assuming two distinct values, also assumes any value between them; hence the image of $  X $
 +
is also an interval on the real line.
  
 
Cauchy's theorem was formulated independently by B. Bolzano (1817) and by A.L. Cauchy (1821).
 
Cauchy's theorem was formulated independently by B. Bolzano (1817) and by A.L. Cauchy (1821).
  
Cauchy's intermediate-value theorem is a generalization of Lagrange's mean-value theorem. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c02099015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c02099016.png" /> are continuous real functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c02099017.png" /> and differentiable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c02099018.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c02099019.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c02099020.png" /> (and therefore <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c02099021.png" />), then there exists a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c02099022.png" /> such that
+
Cauchy's intermediate-value theorem is a generalization of Lagrange's mean-value theorem. If $  f $
 +
and $  g $
 +
are continuous real functions on $  [a, b] $
 +
and differentiable in $  (a, b) $,  
 +
with $  g  ^  \prime  \neq 0 $
 +
on $  (a, b) $(
 +
and therefore $  g (a) \neq g (b) $),  
 +
then there exists a point $  \xi \in (a, b) $
 +
such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c02099023.png" /></td> </tr></table>
+
$$
  
Putting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c02099024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c02099025.png" />, one obtains the ordinary Lagrange mean-value theorem. In geometrical terms, Cauchy's theorem means that on any continuous curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c02099026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c02099027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c02099028.png" />, in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c02099029.png" />-plane having a tangent at each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c02099030.png" />, there exists a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c02099031.png" /> at which the tangent is parallel to the chord connecting the end points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c02099032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c02099033.png" /> of the curve.
+
\frac{f (b) - f (a) }{g (b) - g (a) }
 +
  = \
 +
 
 +
\frac{f ^ { \prime } ( \xi ) }{g  ^  \prime  ( \xi ) }
 +
.
 +
$$
 +
 
 +
Putting $  g (t) = t $,  
 +
$  a \leq  t \leq  b $,  
 +
one obtains the ordinary Lagrange mean-value theorem. In geometrical terms, Cauchy's theorem means that on any continuous curve $  x = f (t) $,  
 +
$  y = g (t) $,  
 +
$  a \leq  t \leq  b $,  
 +
in the $  xy $-
 +
plane having a tangent at each point $  (f (t), g (t)) $,  
 +
there exists a point $  (f ( \xi ), g ( \xi )) $
 +
at which the tangent is parallel to the chord connecting the end points $  (f (a), g (a)) $
 +
and $  (f (b), g (b)) $
 +
of the curve.
  
 
====References====
 
====References====
Line 26: Line 77:
 
The statement in
 
The statement in
  
can be generalized. For continuous real functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c02099034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c02099035.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c02099036.png" /> that are differentiable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c02099037.png" /> there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c02099038.png" /> at which
+
can be generalized. For continuous real functions $  f $
 +
and $  g $
 +
on $  [ a , b ] $
 +
that are differentiable in $  ( a , b ) $
 +
there is an $  x \in ( a , b ) $
 +
at which
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c02099039.png" /></td> </tr></table>
+
$$
 +
[ f (b) - f (a) ] g  ^  \prime  (x)  = \
 +
[ g (b) - g (a) ] f ^ { \prime } (x)
 +
$$
  
 
(cf. [[#References|[a1]]]).
 
(cf. [[#References|[a1]]]).
Line 35: Line 94:
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,  "Principles of mathematical analysis" , McGraw-Hill  (1976)  pp. 107–108</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,  "Principles of mathematical analysis" , McGraw-Hill  (1976)  pp. 107–108</TD></TR></table>
  
Cauchy's theorem in group theory: If the order of a finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c02099040.png" /> is divisible by a prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c02099041.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c02099042.png" /> contains an element of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020990/c02099043.png" />.
+
Cauchy's theorem in group theory: If the order of a finite group $  G $
 +
is divisible by a prime number $  p $,  
 +
then $  G $
 +
contains an element of order $  p $.
  
 
This theorem was first proved by A.L. Cauchy (see [[#References|[1]]]) for permutation groups.
 
This theorem was first proved by A.L. Cauchy (see [[#References|[1]]]) for permutation groups.
Line 43: Line 105:
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Suzuki,  "Group theory" , '''1''' , Springer  (1982)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Suzuki,  "Group theory" , '''1''' , Springer  (1982)</TD></TR></table>

Latest revision as of 15:35, 4 June 2020


Cauchy's theorem on polyhedra: Two closed convex polyhedra are congruent if their true faces, edges and vertices can be put in an incidence-preserving one-to-one correspondence in such a way that corresponding faces are congruent. This is the first theorem about the unique definition of convex surfaces, since the polyhedra of which it speaks are isometric in the sense of an intrinsic metric. The Cauchy theorem is a special case of the theorem stating that every closed convex surface is uniquely defined by its metric (see [4]).

The theorem was first proved by A.L. Cauchy (see [1]).

References

[1] A.L. Cauchy, J. Ecole Polytechnique , 9 (1813) pp. 87–98
[2] A.D. Aleksandrov, "Konvexe Polyeder" , Akademie Verlag (1958) (Translated from Russian)
[3] J. Hadamard, "Géométrie élémentaire" , 2 , Moscow (1957) (In Russian; translated from French)
[4] A.V. Pogorelov, "Unique definition of convex surfaces" Trudy Mat. Inst. Steklov. , 29 (1949) (In Russian)

E.V. Shikin

Cauchy's intermediate-value theorem for continuous functions on closed intervals: Let $ f $ be a continuous real-valued function on $ [a, b] $ and let $ C $ be a number between $ f (a) $ and $ f (b) $. Then there is a point $ \xi \in [a, b] $ such that $ f ( \xi ) = C $. In particular, if $ f (a) $ and $ f (b) $ have different signs, then there is a point $ \xi $ such that $ f ( \xi ) = 0 $. This version of Cauchy's theorem is used to determine intervals in which a function necessarily has zeros. It follows from Cauchy's theorem that the image of an interval on the real line under a continuous mapping into the real line is also an interval. The theorem can be generalized to topological spaces: Any continuous function $ f: X \rightarrow \mathbf R ^ {1} $ defined on a connected topological space $ X $ and assuming two distinct values, also assumes any value between them; hence the image of $ X $ is also an interval on the real line.

Cauchy's theorem was formulated independently by B. Bolzano (1817) and by A.L. Cauchy (1821).

Cauchy's intermediate-value theorem is a generalization of Lagrange's mean-value theorem. If $ f $ and $ g $ are continuous real functions on $ [a, b] $ and differentiable in $ (a, b) $, with $ g ^ \prime \neq 0 $ on $ (a, b) $( and therefore $ g (a) \neq g (b) $), then there exists a point $ \xi \in (a, b) $ such that

$$ \frac{f (b) - f (a) }{g (b) - g (a) } = \ \frac{f ^ { \prime } ( \xi ) }{g ^ \prime ( \xi ) } . $$

Putting $ g (t) = t $, $ a \leq t \leq b $, one obtains the ordinary Lagrange mean-value theorem. In geometrical terms, Cauchy's theorem means that on any continuous curve $ x = f (t) $, $ y = g (t) $, $ a \leq t \leq b $, in the $ xy $- plane having a tangent at each point $ (f (t), g (t)) $, there exists a point $ (f ( \xi ), g ( \xi )) $ at which the tangent is parallel to the chord connecting the end points $ (f (a), g (a)) $ and $ (f (b), g (b)) $ of the curve.

References

[1] V.A. Il'in, E.G. Poznyak, "Fundamentals of mathematical analysis" , 1–2 , MIR (1982) (Translated from Russian)
[2] L.D. Kudryavtsev, "Mathematical analysis" , 1 , Moscow (1973) (In Russian)
[3] S.M. Nikol'skii, "A course of mathematical analysis" , 1–2 , MIR (1977) (Translated from Russian)

L.D. Kudryavtsev

Comments

The statement in

can be generalized. For continuous real functions $ f $ and $ g $ on $ [ a , b ] $ that are differentiable in $ ( a , b ) $ there is an $ x \in ( a , b ) $ at which

$$ [ f (b) - f (a) ] g ^ \prime (x) = \ [ g (b) - g (a) ] f ^ { \prime } (x) $$

(cf. [a1]).

References

[a1] W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1976) pp. 107–108

Cauchy's theorem in group theory: If the order of a finite group $ G $ is divisible by a prime number $ p $, then $ G $ contains an element of order $ p $.

This theorem was first proved by A.L. Cauchy (see [1]) for permutation groups.

References

[1] A.L. Cauchy, "Exercise d'analyse et de physique mathématique" , 3 , Paris (1844) pp. 151–252
[2] A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian)

Comments

References

[a1] M. Suzuki, "Group theory" , 1 , Springer (1982)
How to Cite This Entry:
Cauchy theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cauchy_theorem&oldid=16651
This article was adapted from an original article by E.V. Shikin, L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article