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.

 Starting from the definition of work of a variant force,   where is the force applied to an object along the path defined from the starting point to the ending point . The general form of Newton's second law can be used, . Thus,

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 non-variant force (and/or straight path) when the original time is labeled the initial time and the final time is not labeled.

 by Luis F. Sáez, Ph. D. Comments and Suggestions: LSaez@dallaswinwin.com