Let the unit vector is tangential to the rod. Then the derivative is the curvature to the line: .

The change in a vector due to infinitesimal rotation is the vector product of the rotation vector and the vector itself: , or dividing by

(12.9)

Vector multiplication by gives

or

(12.10)

Since the tangent vector points in the direction of the axis of the rod, . Say the unit vector points to the principal normal: , then we have:

(12.11)

The unit vector is a binormal vector with components so that its magnitude is equal to the curvature . For plane curves, binormal vector is the unit vector normal to the plane. Principal normal is the usual normal to the curve directed towards the center of curvature at that point. Since the elastic energy of the rod is a quadratic function of deformation, in this case it will be function of the components of the vector . However, these terms should not contain the expressions and as they are dependent on the direction of the axis. Finally if we have and axes coinciding with the principal axes of inertia, then we should not have also the term .The total energy of the rod then consists of that due to torsion about the axis and bending about the and axes:

(12.12)

The moment applied about the axis, . Other components of the moment, can similarly be obtained as,

(12.13)

And the elastic energy in terms of the moments are,

(12.14)