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For a proof, imagine two paths 1 and 2, both going from point A to point B. The variation of energy for the particle, taking path 1 from A to B and then path 2 backwards from B to A, is 0; thus, the work is the same in path 1 and 2, i.e., the work is independent of the path followed, as long as it goes from A to B.
For conservative forces, path independence can be interpreted to mean that the work done in going from a point to a point is independent of the moving path chosen (dependent on only the points and ), and that the work done in going around a simple closed loop is :
The work done is given by the dot product of the two vectors, where the result is a scalar. When the force F is constant and the angle θ between the force and the displacement s is also constant, then the work done is given by: = If the force is variable, then work is given by the line integral:
The work per unit of charge is defined as the movement of negligible test charge between two points, and is expressed as the difference in electric potential at those points. The work can be done, for example, by generators , ( electrochemical cells ) or thermocouples generating an electromotive force .
Of this amount, $7600 was spent on the eight round-the-world plane tickets, [13] including two seats on each flight for "Mr. Clock." They flew eastward around the world, ran the clocks side by side for a week, and then flew westward. The crew of each flight helped by supplying the navigational data needed for the comparison with theory.
In electronics, a continuity test is the checking of an electric circuit to see if current flows (that it is in fact a complete circuit). A continuity test is performed by placing a small voltage (wired in series with an LED or noise-producing component such as a piezoelectric speaker ) across the chosen path.
The dependence of work on the path of the thermodynamic process is also unrelated to reversibility, since expansion work, which can be visualized on a pressure–volume diagram as the area beneath the equilibrium curve, is different for different reversible expansion processes (e.g. adiabatic, then isothermal; vs. isothermal, then adiabatic ...
An inexact differential is a differential for which the integral over some two paths with the same end points is different. Specifically, there exist integrable paths ,: [,] such that () = (), () = and In this case, we denote the integrals as | and | respectively to make explicit the path dependence of the change of the quantity we are considering as .