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''of a surface''

The product of the principal curvatures (cf. [[Principal curvature|Principal curvature]]) of a regular surface at a given point.

If

$$ 
\textrm{ I }  =  ds  ^ {2}  = \ 
E  du  ^ {2} + 2 F  du  dv + G  dv  ^ {2}
$$

is the [[First fundamental form|first fundamental form]] of the surface and

$$ 
\textrm{ II }  =  L  du  ^ {2} + 2 M  du  dv + N  dv  ^ {2}
$$

is the [[Second fundamental form|second fundamental form]] of the surface, then the Gaussian curvature can be computed by the formula

$$ 
K  =  
\frac{LN - M  ^ {2} }{EG - F  ^ {2} }
 .
$$

The Gaussian curvature is identical with the Jacobi determinant of the [[Spherical map|spherical map]]:

$$ 
| K | _ {P _ {0}  }  = \ 
\lim\limits _ {d( s) \rightarrow 0 } \ 
{
\frac{S}{s}
 } ,
$$

where  $  P _ {0} $
is a point on the surface,  $  s $
is the area of a domain  $  U $
which contains  $  P _ {0} $, 
$  S $
is the area of the spherical image of  $  U $, 
and  $  d $
is the diameter of the domain. The Gaussian curvature is positive at an [[Elliptic point|elliptic point]], negative at a [[Hyperbolic point|hyperbolic point]], and is zero at a [[Parabolic point|parabolic point]] or a [[Flat point|flat point]]. It may be expressed in terms of the coefficients of the first fundamental form and their derivatives alone (the [[Gauss theorem|Gauss theorem]]), viz.

$$ 
K  =  
\frac{1}{4W  ^ {4} }
 \left |

\begin{array}{ccc}
 E  &E _ {u}  &E _ {v}  \\
 F  &F _ {u}  &F _ {v}  \\
 G  &G _ {u}  &G _ {v}  \\
\end{array}
 \ 
\right | + 
\frac{1}{2W}

\left \{ 
\frac \partial {\partial  u }

\frac{F _ {v} - G _ {u} }{W }
 +

\frac \partial {\partial  v }
 
\frac{F _ {u} - E _ {v} }{W }
 \right \} ,
$$

where

$$ 
W  ^ {2}  =  EG - F  ^ {2} .
$$

Since the Gaussian curvature depends on the metric only, i.e. on the coefficients of the first fundamental form, the Gaussian curvature is invariant under isometric deformation (cf. [[Deformation, isometric|Deformation, isometric]]). The Gaussian curvature plays a special role in the theory of surfaces, and many formulas are available for its computation, [[#References|[2]]].

The concept was introduced by C.F. Gauss [[#References|[1]]], and was named after him.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C.F. Gauss,   "Allgemeine Flächentheorie" , W. Engelmann , Leipzig  (1900)  (Translated from Latin)  {{MR|}}  {{ZBL|31.0599.11}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Reichardt,   "Einführung in die Differentialgeometrie" , Springer  (1960)  {{MR|0116267}} {{ZBL|0091.34001}} </TD></TR></table>

====Comments====
The total Gaussian curvature (often also abbreviated to total curvature) is the quantity

$$ 
\int\limits \int\limits K  d \sigma .
$$

(See also [[Gauss–Bonnet theorem|Gauss–Bonnet theorem]].)

For a smooth space curve  $  C $
given by  $  x = x ( s) $, 
the total curvature  $  K $
of  $  C $
is defined as the length of the spherical image of  $  C $(
cf. also [[Spherical indicatrix|Spherical indicatrix]]) and can be expressed using the [[Frénet formulas|Frénet formulas]]  $  e _ {1}  ^  \prime  = \kappa _ {1} e _ {2} $, 
$  e _ {2}  ^  \prime  = - \kappa _ {1} e _ {1} + \kappa _ {2} e _ {3} $, 
$  e _ {3}  ^  \prime  = - \kappa _ {2} e _ {2} $
for a Frénet frame (cf. [[Frénet trihedron|Frénet trihedron]])  $  ( x, e _ {1} , e _ {2} , e _ {3} ) $
along  $  C $
by

$$ 
K  =  \int\limits \kappa _ {1}  ds.
$$