The term "total curvature" is used in two different ways in differential geometry.
The total curvature, also called the third curvature, of a space curve with line elements 
, 
, and 
along the normal, tangent, and binormal vectors respectively,
is defined as the quantity
 |  |  |
(1)
|
 |  |  |
(2)
|
where 
is the curvature and 
is the torsion (Kreyszig 1991,
p. 47). The term is apparently also applied to the derivative directly 
, namely
 |
(3)
|
(Kreyszig 1991, p. 47).
The second use of "total curvature" is as a synonym for Gaussian curvature (Kreyszig 1991, p. 131).
