Namespaces
Variants
Actions

Difference between revisions of "Direction field"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (OldImage template added)
 
(2 intermediate revisions by 2 users not shown)
Line 1: Line 1:
 +
<!--
 +
d0327801.png
 +
$#A+1 = 30 n = 0
 +
$#C+1 = 30 : ~/encyclopedia/old_files/data/D032/D.0302780 Direction field
 +
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}}
 +
 
A geometrical interpretation of the set of line elements corresponding to a system of ordinary differential equations
 
A geometrical interpretation of the set of line elements corresponding to a system of ordinary differential equations
  
<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/d/d032/d032780/d0327801.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\dot{x} _ {i}  = \
 +
f _ {i} ( t, x _ {1}, \dots, x _ {n} ),\ \
 +
i = 1, \dots, n.
 +
$$
  
 
A line element is defined as a sequence of numbers
 
A line element is defined as a sequence of numbers
  
<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/d/d032/d032780/d0327802.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
t, x _ {1}, \dots, x _ {n} ,
 +
$$
  
<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/d/d032/d032780/d0327803.png" /></td> </tr></table>
+
$$
 +
f _ {1} ( t, x _ {1}, \dots, x _ {n} ), \dots, f _ {n} ( t, x _ {1}, \dots, x _ {n} ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032780/d0327804.png" /> is a point of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032780/d0327805.png" />, at which the terms on the right-hand side of (1) are defined. A line element (2) can be described as a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032780/d0327806.png" /> together with the direction with direction cosines
+
where $  ( t, x _ {1}, \dots, x _ {n} ) $
 +
is a point of the domain $  G \subset  \mathbf R ^ {n + 1 } $,  
 +
at which the terms on the right-hand side of (1) are defined. A line element (2) can be described as a point $  ( t, x _ {1}, \dots, x _ {n} ) \in G $
 +
together with the direction with direction cosines
  
<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/d/d032/d032780/d0327807.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
\left ( 1 +
 +
\sum _ {k = 1 } ^ { n }  f _ {k} ^ { 2 }
 +
\right )  ^ {-1/2} ,\ \
 +
f _ {i} \left ( 1 +
 +
\sum _ {k = 1 } ^ { n }  f _ {k} ^ { 2 }
 +
\right ) ^ {-1/2} ,\ \
 +
i = 1, \dots, n,
 +
$$
  
 
which is represented by a short segment through the point, parallel to the vector
 
which is represented by a short segment through the point, parallel to the vector
  
<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/d/d032/d032780/d0327808.png" /></td> </tr></table>
+
$$
 +
( 1, f _ {1} ( t, x _ {1}, \dots, x _ {n} ), \dots,
 +
f _ {n} ( t, x _ {1}, \dots, x _ {n} )).
 +
$$
  
 
For a system in symmetric form,
 
For a system in symmetric form,
  
<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/d/d032/d032780/d0327809.png" /></td> </tr></table>
+
$$
  
unlike system (1), the field may also contain directions orthogonal to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032780/d03278010.png" />-axis.
+
\frac{dt}{f _ {0} ( t, x _ {1}, \dots, x _ {n} ) }
 +
  = \dots = \
  
Consider any [[Integral curve|integral curve]] of the system (1). At every point of this curve, the direction field corresponding to the point is tangent to the curve at that point; any curve with this property is an integral curve of (1). Thus, the specification of a direction field is equivalent to the specification of the system (1), and the problem of integrating this system amounts to the determination of the curves in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032780/d03278011.png" /> the tangents of which at each point possess the directions defined by the formulas (3), i.e. the directions coinciding with the directions of the field at that point.
+
\frac{dx _ {n} }{f _ {n} ( t, x _ {1}, \dots, x _ {n} ) }
 +
,
 +
$$
  
The geometrical interpretation is particularly easy to visualize when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032780/d03278012.png" />. In that case, for each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032780/d03278013.png" /> of the domain of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032780/d03278014.png" /> of the right-hand side in the first-order equation
+
unlike system (1), the field may also contain directions orthogonal to the $  t $-axis.
  
<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/d/d032/d032780/d03278015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
Consider any [[Integral curve|integral curve]] of the system (1). At every point of this curve, the direction field corresponding to the point is tangent to the curve at that point; any curve with this property is an integral curve of (1). Thus, the specification of a direction field is equivalent to the specification of the system (1), and the problem of integrating this system amounts to the determination of the curves in  $  \mathbf R ^ {n + 1 } $
 +
the tangents of which at each point possess the directions defined by the formulas (3), i.e. the directions coinciding with the directions of the field at that point.
  
there exists a short segment through the point, with slope <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032780/d03278016.png" />, such that the (directed) angle between the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032780/d03278017.png" />-axis and the segment is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032780/d03278018.png" /> (see Fig.).
+
The geometrical interpretation is particularly easy to visualize when  $  n = 1 $.
 +
In that case, for each point  $  ( t, x ) $
 +
of the domain of definition  $  G \subset  \mathbf R  ^ {2} $
 +
of the right-hand side in the first-order equation
 +
 
 +
$$ \tag{4 }
 +
 
 +
\frac{dx }{dt }
 +
  = \
 +
f ( t, x)
 +
$$
 +
 
 +
there exists a short segment through the point, with slope $  f ( t, x) $,  
 +
such that the (directed) angle between the $  t $-axis and the segment is equal to $  { \mathop{\rm arc}  \mathop{\rm tan} }  f ( t, x) $ (see Fig.).
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d032780a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/d032780a.gif" />
Line 37: Line 88:
 
Frequently, one considers equation (4) together with the differential equation
 
Frequently, one considers equation (4) together with the differential equation
  
<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/d/d032/d032780/d03278019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
$$ \tag{5 }
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032780/d03278020.png" /> for points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032780/d03278021.png" /> at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032780/d03278022.png" />, defining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032780/d03278023.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032780/d03278024.png" />, provided this definition preserves the continuity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032780/d03278025.png" />. By this device, the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032780/d03278026.png" /> for the pair of equations (4), (5) is extended to a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032780/d03278027.png" /> by the addition of the points at which the direction is parallel to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032780/d03278028.png" />-axis, and the integral curves are also allowed to have points with vertical tangents.
+
\frac{dt }{dx }
 +
  = F ( t, x),
 +
$$
  
If the direction field is drawn in sufficient detail in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032780/d03278029.png" /> for equation (4) (or in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032780/d03278030.png" /> for the pair of equations (4), (5)), the pattern of line segments will provide a rough qualitative idea of the behaviour of the integral curves. This idea is the basis for the approximate graphical method for solving equation (4), known as the method of isoclines, in which the direction field is constructed with the aid of isoclines (cf. [[Isocline|Isocline]]). The geometrical relationship between the direction field and the integral curves is also the basis for an approximate numerical method for solving equation (4) — the [[Euler method|Euler method]].
+
where  $  F ( t, x) = 1/f ( t, x) $
 +
for points  $  ( t, x) \in G $
 +
at which  $  f ( t, x) \neq 0 $,
 +
defining  $  F ( t, x) = 0 $
 +
for  $  ( t, x) \in \mathbf R  ^ {2} \setminus  G $,
 +
provided this definition preserves the continuity of  $  F ( t, x) $.
 +
By this device, the domain  $  G $
 +
for the pair of equations (4), (5) is extended to a domain  $  G _ {0} $
 +
by the addition of the points at which the direction is parallel to the  $  x $-axis, and the integral curves are also allowed to have points with vertical tangents.
 +
 
 +
If the direction field is drawn in sufficient detail in $  G $
 +
for equation (4) (or in $  G _ {0} $
 +
for the pair of equations (4), (5)), the pattern of line segments will provide a rough qualitative idea of the behaviour of the integral curves. This idea is the basis for the approximate graphical method for solving equation (4), known as the method of isoclines, in which the direction field is constructed with the aid of isoclines (cf. [[Isocline|Isocline]]). The geometrical relationship between the direction field and the integral curves is also the basis for an approximate numerical method for solving equation (4) — the [[Euler method|Euler method]].
  
 
For autonomous systems (cf. [[Autonomous system|Autonomous system]]) of ordinary differential equations there is a more convenient and intuitive geometrical interpretation in terms of a [[Vector field|vector field]] (cf. also [[Vector field on a manifold|Vector field on a manifold]]) — the field of phase velocities in the phase space of the system.
 
For autonomous systems (cf. [[Autonomous system|Autonomous system]]) of ordinary differential equations there is a more convenient and intuitive geometrical interpretation in terms of a [[Vector field|vector field]] (cf. also [[Vector field on a manifold|Vector field on a manifold]]) — the field of phase velocities in the phase space of the system.
Line 47: Line 112:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.S. Pontryagin,  "Ordinary differential equations" , Addison-Wesley  (1962)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Kamke,  "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''1. Gewöhnliche Differentialgleichungen''' , Chelsea, reprint  (1947)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G. Sansone,  "Ordinary differential equations" , '''1–2''' , Zanichelli  (1948–1949)  (In Italian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.S. Pontryagin,  "Ordinary differential equations" , Addison-Wesley  (1962)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Kamke,  "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''1. Gewöhnliche Differentialgleichungen''' , Chelsea, reprint  (1947)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G. Sansone,  "Ordinary differential equations" , '''1–2''' , Zanichelli  (1948–1949)  (In Italian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
The direction field, or line elements field, is often used in textbooks on differential equations in relation with applications in mechanics, biology and econometrics.
 
The direction field, or line elements field, is often used in textbooks on differential equations in relation with applications in mechanics, biology and econometrics.
 +
 +
{{OldImage}}

Latest revision as of 11:49, 26 March 2023


A geometrical interpretation of the set of line elements corresponding to a system of ordinary differential equations

$$ \tag{1 } \dot{x} _ {i} = \ f _ {i} ( t, x _ {1}, \dots, x _ {n} ),\ \ i = 1, \dots, n. $$

A line element is defined as a sequence of numbers

$$ \tag{2 } t, x _ {1}, \dots, x _ {n} , $$

$$ f _ {1} ( t, x _ {1}, \dots, x _ {n} ), \dots, f _ {n} ( t, x _ {1}, \dots, x _ {n} ), $$

where $ ( t, x _ {1}, \dots, x _ {n} ) $ is a point of the domain $ G \subset \mathbf R ^ {n + 1 } $, at which the terms on the right-hand side of (1) are defined. A line element (2) can be described as a point $ ( t, x _ {1}, \dots, x _ {n} ) \in G $ together with the direction with direction cosines

$$ \tag{3 } \left ( 1 + \sum _ {k = 1 } ^ { n } f _ {k} ^ { 2 } \right ) ^ {-1/2} ,\ \ f _ {i} \left ( 1 + \sum _ {k = 1 } ^ { n } f _ {k} ^ { 2 } \right ) ^ {-1/2} ,\ \ i = 1, \dots, n, $$

which is represented by a short segment through the point, parallel to the vector

$$ ( 1, f _ {1} ( t, x _ {1}, \dots, x _ {n} ), \dots, f _ {n} ( t, x _ {1}, \dots, x _ {n} )). $$

For a system in symmetric form,

$$ \frac{dt}{f _ {0} ( t, x _ {1}, \dots, x _ {n} ) } = \dots = \ \frac{dx _ {n} }{f _ {n} ( t, x _ {1}, \dots, x _ {n} ) } , $$

unlike system (1), the field may also contain directions orthogonal to the $ t $-axis.

Consider any integral curve of the system (1). At every point of this curve, the direction field corresponding to the point is tangent to the curve at that point; any curve with this property is an integral curve of (1). Thus, the specification of a direction field is equivalent to the specification of the system (1), and the problem of integrating this system amounts to the determination of the curves in $ \mathbf R ^ {n + 1 } $ the tangents of which at each point possess the directions defined by the formulas (3), i.e. the directions coinciding with the directions of the field at that point.

The geometrical interpretation is particularly easy to visualize when $ n = 1 $. In that case, for each point $ ( t, x ) $ of the domain of definition $ G \subset \mathbf R ^ {2} $ of the right-hand side in the first-order equation

$$ \tag{4 } \frac{dx }{dt } = \ f ( t, x) $$

there exists a short segment through the point, with slope $ f ( t, x) $, such that the (directed) angle between the $ t $-axis and the segment is equal to $ { \mathop{\rm arc} \mathop{\rm tan} } f ( t, x) $ (see Fig.).

Figure: d032780a

Frequently, one considers equation (4) together with the differential equation

$$ \tag{5 } \frac{dt }{dx } = F ( t, x), $$

where $ F ( t, x) = 1/f ( t, x) $ for points $ ( t, x) \in G $ at which $ f ( t, x) \neq 0 $, defining $ F ( t, x) = 0 $ for $ ( t, x) \in \mathbf R ^ {2} \setminus G $, provided this definition preserves the continuity of $ F ( t, x) $. By this device, the domain $ G $ for the pair of equations (4), (5) is extended to a domain $ G _ {0} $ by the addition of the points at which the direction is parallel to the $ x $-axis, and the integral curves are also allowed to have points with vertical tangents.

If the direction field is drawn in sufficient detail in $ G $ for equation (4) (or in $ G _ {0} $ for the pair of equations (4), (5)), the pattern of line segments will provide a rough qualitative idea of the behaviour of the integral curves. This idea is the basis for the approximate graphical method for solving equation (4), known as the method of isoclines, in which the direction field is constructed with the aid of isoclines (cf. Isocline). The geometrical relationship between the direction field and the integral curves is also the basis for an approximate numerical method for solving equation (4) — the Euler method.

For autonomous systems (cf. Autonomous system) of ordinary differential equations there is a more convenient and intuitive geometrical interpretation in terms of a vector field (cf. also Vector field on a manifold) — the field of phase velocities in the phase space of the system.

References

[1] L.S. Pontryagin, "Ordinary differential equations" , Addison-Wesley (1962) (Translated from Russian)
[2] E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 1. Gewöhnliche Differentialgleichungen , Chelsea, reprint (1947)
[3] G. Sansone, "Ordinary differential equations" , 1–2 , Zanichelli (1948–1949) (In Italian)

Comments

The direction field, or line elements field, is often used in textbooks on differential equations in relation with applications in mechanics, biology and econometrics.


🛠️ This page contains images that should be replaced by better images in the SVG file format. 🛠️
How to Cite This Entry:
Direction field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Direction_field&oldid=12439
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article