Namespaces
Variants
Actions

Difference between revisions of "Cylinder coordinates"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
 +
<!--
 +
c0276001.png
 +
$#A+1 = 34 n = 0
 +
$#C+1 = 34 : ~/encyclopedia/old_files/data/C027/C.0207600 Cylinder coordinates,
 +
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}}
 +
 
''cylindrical coordinates''
 
''cylindrical coordinates''
  
Numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027600/c0276001.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027600/c0276002.png" /> connected with the Cartesian coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027600/c0276003.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027600/c0276004.png" /> by the formulas:
+
Numbers $  \rho , \phi $
 +
and $  z $
 +
connected with the Cartesian coordinates $  x, y $
 +
and $  z $
 +
by the formulas:
  
<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/c027/c027600/c0276005.png" /></td> </tr></table>
+
$$
 +
= \rho  \cos  \phi ,\ \
 +
= \rho  \sin  \phi ,\ \
 +
= z,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027600/c0276006.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027600/c0276007.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027600/c0276008.png" />. The coordinate surfaces (see Fig.) are: circular cylinders <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027600/c0276009.png" />, half-planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027600/c02760010.png" /> and planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027600/c02760011.png" />.
+
where $  0 \leq  \rho < \infty $,
 +
0 \leq  \phi < 2 \pi $,  
 +
$  - \infty < z < \infty $.  
 +
The coordinate surfaces (see Fig.) are: circular cylinders $  ( \rho = \textrm{ const } ) $,  
 +
half-planes $  ( \phi = \textrm{ const } ) $
 +
and planes $  ( z = \textrm{ const } ) $.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c027600a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c027600a.gif" />
Line 15: Line 40:
 
The [[Lamé coefficients|Lamé coefficients]] are:
 
The [[Lamé coefficients|Lamé coefficients]] are:
  
<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/c027/c027600/c02760012.png" /></td> </tr></table>
+
$$
 +
L _  \rho  = \
 +
L _ {z}  = 1,\ \
 +
L _  \phi  = \rho .
 +
$$
  
 
The area element of a surface is:
 
The area element of a surface is:
  
<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/c027/c027600/c02760013.png" /></td> </tr></table>
+
$$
 +
ds  = \
 +
\sqrt {\rho  ^ {2} ( d \rho  d \phi )  ^ {2} +
 +
( d \rho  dz)  ^ {2} +
 +
\rho  ^ {2} ( d \phi  dz)  ^ {2} } .
 +
$$
  
 
The volume element is:
 
The volume element is:
  
<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/c027/c027600/c02760014.png" /></td> </tr></table>
+
$$
 +
dV  = \rho  d \rho  d \phi  dz.
 +
$$
  
 
The differentiation operations of vector analysis are given by:
 
The differentiation operations of vector analysis are given by:
  
<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/c027/c027600/c02760015.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm grad} _  \rho  f  = \
  
<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/c027/c027600/c02760016.png" /></td> </tr></table>
+
\frac{\partial  f }{\partial  \rho }
 +
,\ \
 +
\mathop{\rm grad} _  \phi  f  = \
 +
{
 +
\frac{1} \rho
 +
}
  
<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/c027/c027600/c02760017.png" /></td> </tr></table>
+
\frac{\partial  f }{\partial  \phi }
 +
,\ \
 +
\mathop{\rm grad} _ {z}  f  = \
  
<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/c027/c027600/c02760018.png" /></td> </tr></table>
+
\frac{\partial  f }{\partial  z }
 +
;
 +
$$
  
<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/c027/c027600/c02760019.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm div}  \mathbf a  = {
 +
\frac{1} \rho
 +
} a _  \rho  +
 +
\frac{\partial  a _  \rho  }{\partial  \rho }
 +
+ {
 +
\frac{1} \rho
 +
}
 +
\frac{\partial  a _  \phi  }{\partial  \phi }
 +
+
 +
\frac{\partial  a _ {z} }{\partial  z }
 +
;
 +
$$
  
Generalized cylinder coordinates are numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027600/c02760020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027600/c02760021.png" /> connected with Cartesian coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027600/c02760022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027600/c02760023.png" /> by the formulas
+
$$
 +
\mathop{\rm curl} _  \rho  \mathbf a  = {
 +
\frac{1} \rho
 +
}
 +
\frac{\partial  a _ {z} }{\partial  \phi }
 +
-
 +
\frac{\partial  a _  \phi  }{\partial  z }
 +
,\  \mathop{\rm curl} _  \phi  \mathbf a  = \
  
<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/c027/c027600/c02760024.png" /></td> </tr></table>
+
\frac{\partial  a _  \rho  }{\partial  z }
 +
-  
 +
\frac{\partial  a _ {z} }{\partial  \rho }
 +
;
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027600/c02760025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027600/c02760026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027600/c02760027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027600/c02760028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027600/c02760029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027600/c02760030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027600/c02760031.png" />. The coordinate surfaces are: elliptic cylinders <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027600/c02760032.png" />, half-planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027600/c02760033.png" /> and planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027600/c02760034.png" />.
+
$$
 +
\mathop{\rm curl} _ {z}  \mathbf a  = {
 +
\frac{1} \rho
 +
} a _  \phi  +
 +
\frac{\partial  a _  \phi  }{\partial  \rho }
 +
- {
 +
\frac{1} \rho
 +
}
 +
\frac{\partial  a _  \rho  }{\partial  \phi }
 +
;
 +
$$
  
 +
$$
 +
\Delta f  = 
 +
\frac{\partial  ^ {2} f }{\partial  \rho  ^ {2} }
  
 +
+ {
 +
\frac{1} \rho
 +
}
 +
\frac{\partial  f }{\partial  \rho }
 +
+ {
 +
\frac{1}{
 +
\rho  ^ {2} }
 +
}
 +
\frac{\partial  ^ {2} f }{\partial  \phi  ^ {2}
 +
}
 +
+
 +
\frac{\partial  ^ {2} \phi }{\partial  z  ^ {2} }
 +
.
 +
$$
  
====Comments====
+
Generalized cylinder coordinates are numbers  $  u , v $
 +
and  $  w $
 +
connected with Cartesian coordinates  $  x, y $
 +
and  $  z $
 +
by the formulas
  
 +
$$
 +
x  =  au  \cos  v ,\ \
 +
y  =  bu  \sin  v ,\ \
 +
z  =  cw,
 +
$$
 +
 +
where  $  0 \leq  u < \infty $,
 +
$  0 \leq  v < 2 \pi $,
 +
$  - \infty < w < \infty $,
 +
$  a > 0 $,
 +
$  b > 0 $,
 +
$  c > 0 $,
 +
$  a \neq b $.
 +
The coordinate surfaces are: elliptic cylinders  $  ( u = \textrm{ const } ) $,
 +
half-planes  $  ( v = \textrm{ const } ) $
 +
and planes  $  ( w = \textrm{ const } ) $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  U.G. Chambers,  "A course in vector analysis" , Chapman &amp; Hall  (1969)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  U.G. Chambers,  "A course in vector analysis" , Chapman &amp; Hall  (1969)</TD></TR></table>
 +
 +
 +
{{OldImage}}

Latest revision as of 11:19, 26 March 2023


cylindrical coordinates

Numbers $ \rho , \phi $ and $ z $ connected with the Cartesian coordinates $ x, y $ and $ z $ by the formulas:

$$ x = \rho \cos \phi ,\ \ y = \rho \sin \phi ,\ \ z = z, $$

where $ 0 \leq \rho < \infty $, $ 0 \leq \phi < 2 \pi $, $ - \infty < z < \infty $. The coordinate surfaces (see Fig.) are: circular cylinders $ ( \rho = \textrm{ const } ) $, half-planes $ ( \phi = \textrm{ const } ) $ and planes $ ( z = \textrm{ const } ) $.

Figure: c027600a

The system of cylinder coordinates is orthogonal.

The Lamé coefficients are:

$$ L _ \rho = \ L _ {z} = 1,\ \ L _ \phi = \rho . $$

The area element of a surface is:

$$ ds = \ \sqrt {\rho ^ {2} ( d \rho d \phi ) ^ {2} + ( d \rho dz) ^ {2} + \rho ^ {2} ( d \phi dz) ^ {2} } . $$

The volume element is:

$$ dV = \rho d \rho d \phi dz. $$

The differentiation operations of vector analysis are given by:

$$ \mathop{\rm grad} _ \rho f = \ \frac{\partial f }{\partial \rho } ,\ \ \mathop{\rm grad} _ \phi f = \ { \frac{1} \rho } \frac{\partial f }{\partial \phi } ,\ \ \mathop{\rm grad} _ {z} f = \ \frac{\partial f }{\partial z } ; $$

$$ \mathop{\rm div} \mathbf a = { \frac{1} \rho } a _ \rho + \frac{\partial a _ \rho }{\partial \rho } + { \frac{1} \rho } \frac{\partial a _ \phi }{\partial \phi } + \frac{\partial a _ {z} }{\partial z } ; $$

$$ \mathop{\rm curl} _ \rho \mathbf a = { \frac{1} \rho } \frac{\partial a _ {z} }{\partial \phi } - \frac{\partial a _ \phi }{\partial z } ,\ \mathop{\rm curl} _ \phi \mathbf a = \ \frac{\partial a _ \rho }{\partial z } - \frac{\partial a _ {z} }{\partial \rho } ; $$

$$ \mathop{\rm curl} _ {z} \mathbf a = { \frac{1} \rho } a _ \phi + \frac{\partial a _ \phi }{\partial \rho } - { \frac{1} \rho } \frac{\partial a _ \rho }{\partial \phi } ; $$

$$ \Delta f = \frac{\partial ^ {2} f }{\partial \rho ^ {2} } + { \frac{1} \rho } \frac{\partial f }{\partial \rho } + { \frac{1}{ \rho ^ {2} } } \frac{\partial ^ {2} f }{\partial \phi ^ {2} } + \frac{\partial ^ {2} \phi }{\partial z ^ {2} } . $$

Generalized cylinder coordinates are numbers $ u , v $ and $ w $ connected with Cartesian coordinates $ x, y $ and $ z $ by the formulas

$$ x = au \cos v ,\ \ y = bu \sin v ,\ \ z = cw, $$

where $ 0 \leq u < \infty $, $ 0 \leq v < 2 \pi $, $ - \infty < w < \infty $, $ a > 0 $, $ b > 0 $, $ c > 0 $, $ a \neq b $. The coordinate surfaces are: elliptic cylinders $ ( u = \textrm{ const } ) $, half-planes $ ( v = \textrm{ const } ) $ and planes $ ( w = \textrm{ const } ) $.

References

[a1] U.G. Chambers, "A course in vector analysis" , Chapman & Hall (1969)



🛠️ This page contains images that should be replaced by better images in the SVG file format. 🛠️
How to Cite This Entry:
Cylinder coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cylinder_coordinates&oldid=16846
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article