The metric is expressed in the + sign convention.

Using either sign convention for work, the change in internal energy of the system is:

The sign convention can be established by rigorous attention to coordinate systems.

Note however that in areas of optics other than design, other sign conventions are sometimes used.

Nevertheless, for historical reasons, the customary sign convention is to consider work done by the system on its surroundings as positive.

This is the sign convention for work in many textbooks on physics.

The formulas hold for either sign convention, unless otherwise noted.

The sign convention is such that positive work is performed by the system on the environment.

For the bending moment diagram the normal sign convention was used.

See section 6.7 for a discussion of the decomposition (but note different sign conventions).