This section is suitable for the reader familiar with calculus. Skipping this section does not break the continuity of the analysis of energy presented in this notes.
where is the velocity of the object, is the initial position of the object at the time , and is the final position of the object at the time . Thus, . Considering the derivative of and the fact that the scalar product of vectors is commutative, the previous derivative is the same as the argument of the integral; that is, where the derivative of the integral is the same as the argument of the integral
where is the velocity of the object at the time and is the velocity of the object at the time . The previous result is the same as the result obtained for the case of a nonvariant force (and/or straight path) when the original time is labeled the initial time and the final time is not labeled.

