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).