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''of a first-order partial differential equation''

A family

$$ 
x  =  x ( t),\ \ 
u  =  y ( t),\ \ 
u _ {x}  =  p ( t)
$$

of continuously-differentiable functions in an interval  $  \alpha < t < \beta $, 
satisfying the equations

$$ 
x  ^  \prime  ( t)  =  F _ {p} ,\ \ 
y  ^  \prime  ( t)  =  pF _ {p} ,\ \ 
p  ^  \prime  ( t)  =  - F _ {x} - p F _ {y} ,
$$

where the multiplication of the vectors is the scalar product, and where

$$ \tag{* }
F ( x, u , u _ {x} )  =  0
$$

is a non-linear first-order partial differential equation in the unknown function  $  u:  \Omega \subseteq \mathbf R  ^ {n} \rightarrow \mathbf R $. 
Here  $  u _ {x} =  \mathop{\rm grad}  u $, 
$  F ( x, y, p):  \Omega \times \mathbf R \times \mathbf R  ^ {n} \rightarrow \mathbf R $, 
$  x, p \in \mathbf R  ^ {n} $, 
$  y \in \mathbf R $, 
$  n \in \mathbf N $.

The importance of a characteristic strip consists in the fact that it is used in the study of, and in the search for, solutions of equation (*).

See also [[Characteristic|Characteristic]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Kamke,   "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''2. Partielle Differentialgleichungen erster Ordnung für die gesuchte Funktion''' , Akad. Verlagsgesell.  (1944)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P. Hartman,   "Ordinary differential equations" , Birkhäuser  (1982)</TD></TR></table>

====Comments====
A characteristic strip is sometimes called a bicharacteristic.

In the modern theory, the characteristic strips of a partial differential equation carry the [[Wave front|wave front]] sets of solutions of a partial differential equation.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Courant,   D. Hilbert,   "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience  (1962)  (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.V. Hörmander,   "The analysis of linear partial differential operators" , '''1''' , Springer  (1983)</TD></TR></table>