Path-independent, line integral

This statement means two things. First, if the line integral / is independent of path, then dM/dy = dN/dx. Second, if dM/dy = dN/dx, then the line integral / is independent of path. The equality given by Eq. (14-3) is commonly called the Maxwell relationship. 

This is one of the well-known Maxwell relationships. Another useful property of functions whose line integrals are independent of path is the following one. If the line integral / is independent of path, a function /(x, y) exists such that... 

If we calculate the above line integral for an arbitrary approximate 1 the result will, in general, be dependent on the path y used. However path independence is guaranteed [60] if v c satisfies the condition... 

One should note that the potential function specified by Eq. (1.5.12) is actually a mathematical construct reminiscent of the concept of potential energy, commonly encountered in elementary treatments. However, the two ideas should not be confused the potential energy is often invoked because, as shown below, the integrand does arise in the development of the concept of energy. The potential 0, however, can be invoked only for line integrals that are independent of the path, i.e., that vanish when taken around a loop. 

Path independent energy line integral around a crack tip... 

An integral of a differential with several independent variables is a line integral, carried out on a specified path in the space of the independent variables. 

Theorem 1 If dF is an exact differential, then the line integral J dF depends only on the initial and final points and not on the choice of curve joining these points. Further, the line integral equals the value of the function F at the final point minus the value of the function at the beginning point. We say that the line integral of an exact differential is path-independent... 

The line integral of an exact differential form depends only on the initial and final points of integration and is independent of the path of integration. This can be easily seen by substituting Eq. (A-31) into Eq. (A-33). The result is... 

The extension of the concept of integration considered in this section involves continuous summation of a differential expression along a specified path C. For the case of two independent variables, the line integral can be defined as follows ... 

Under certain conditions, the line integral in Eq. 9.55 is independent of path. Suppose both integrals on the right-hand side are exact, so for example, we could write... 

Calculation of the radiative-recoil correction generated by the one-loop polarization insertions in the exchanged photon lines in Fig. 5.2 follows the same path as calculation of the correction induced by the insertions in the electron line. The respective correction was independently calculated analytically both in the skeleton integral approach [8] and with the help of the Braun formula... 

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