Difference between revisions of "Curvilinear coordinates"

From Glossary of Meteorology
imported>Perlwikibot
(Created page with " {{TermHeader}} {{TermSearch}} <div class="termentry"> <div class="term"> == curvilinear coordinates == </div> <div class="definition"><div class="short_definition">Any...")
 
imported>Perlwikibot
 
Line 9: Line 9:
 
   </div>
 
   </div>
  
<div class="definition"><div class="short_definition">Any [[linear]] coordinates that are not [[Cartesian coordinates]].</div><br/> <div class="paragraph">If ''u'', ''v'', ''w'' are three functions of the Cartesian coordinates ''x'', ''y'', ''z'' and if at least one of these  functions is not a linear combination of ''x'', ''y'', ''z'', then ''u'', ''v'', ''w'' are curvilinear coordinates of the  point the Cartesian coordinates of which are ''x'', ''y'', ''z'' provided that the [[Jacobian]] &part;(''u'',''v'',''w'')/&part;(''x'',''y'',''z'')  is not equal to zero. Any surface along which one of the three curvilinear coordinates is constant  is called a [[coordinate surface]]; there are three families of such surfaces. Any line along which two  of the three curvilinear coordinates are constant is called a [[coordinate line]]; there are three sets  of such lines. Three distinct coordinate lines may be drawn through each point of space. The three  straight lines each of which is tangent to one of the coordinate lines at a given point in space are  called the local axes. If the local axes are everywhere mutually perpendicular, the curvilinear coordinates  are said to be [[orthogonal]] or rectangular. Examples of frequently used curvilinear coordinates  are [[polar coordinates]] and [[cylindrical coordinates]]. <br/>''See also'' [[natural coordinates]], [[spherical  coordinates]].</div><br/> </div>
+
<div class="definition"><div class="short_definition">Any [[linear]] coordinates that are not [[Cartesian coordinates]].</div><br/> <div class="paragraph">If ''u'', ''v'', ''w'' are three functions of the Cartesian coordinates ''x'', ''y'', ''z'' and if at least one of these  functions is not a linear combination of ''x'', ''y'', ''z'', then ''u'', ''v'', ''w'' are curvilinear coordinates of the  point the Cartesian coordinates of which are ''x'', ''y'', ''z'' provided that the [[Jacobian]] &part;(''u'',''v'',''w'')/&part;(''x'',''y'',''z'')  is not equal to zero. Any surface along which one of the three curvilinear coordinates is constant  is called a [[coordinate surface]]; there are three families of such surfaces. Any line along which two  of the three curvilinear coordinates are constant is called a [[coordinate line]]; there are three sets  of such lines. Three distinct coordinate lines may be drawn through each point of space. The three  straight lines each of which is tangent to one of the coordinate lines at a given point in space are  called the local axes. If the local axes are everywhere mutually perpendicular, the curvilinear coordinates  are said to be [[orthogonal]] or rectangular. Examples of frequently used curvilinear coordinates  are [[polar coordinates]] and [[cylindrical coordinates]]. <br/>''See also'' [[natural coordinates]], [[spherical coordinates|spherical  coordinates]].</div><br/> </div>
 
</div>
 
</div>
  

Latest revision as of 15:45, 25 April 2012



curvilinear coordinates

Any linear coordinates that are not Cartesian coordinates.

If u, v, w are three functions of the Cartesian coordinates x, y, z and if at least one of these functions is not a linear combination of x, y, z, then u, v, w are curvilinear coordinates of the point the Cartesian coordinates of which are x, y, z provided that the Jacobian ∂(u,v,w)/∂(x,y,z) is not equal to zero. Any surface along which one of the three curvilinear coordinates is constant is called a coordinate surface; there are three families of such surfaces. Any line along which two of the three curvilinear coordinates are constant is called a coordinate line; there are three sets of such lines. Three distinct coordinate lines may be drawn through each point of space. The three straight lines each of which is tangent to one of the coordinate lines at a given point in space are called the local axes. If the local axes are everywhere mutually perpendicular, the curvilinear coordinates are said to be orthogonal or rectangular. Examples of frequently used curvilinear coordinates are polar coordinates and cylindrical coordinates.
See also natural coordinates, spherical coordinates.