A thermal elastic plastic coupled material model and an advanced method for calculating the incremental path independent integral are introduced.

By using a complex function method and changing J integral to complex form,the path independence of J integral near I mode, Ⅱ mode and mixed mode crack tips in principal elasticity direction were proved, and the computing formulae of the J integral were derived.

And then,by calculating the mean values of the unwrapped phase along multiple certain directions,the goal of path independence can be achieved.

The problem of path independence of choice function is investigated in this paper.

Using the canonical transformation and the method of path integrals, the quantum wavefunction of the time-dependent RLC circuit after quantization is solved, and the quantum fluctuations of the charge and current are investigated.

The mathematical structure and physical sense of Feynman s path integrals have been redefined,by using the theory of stochastic processes.

Using the canonical transformation and the method of path integrals,the exact wavefunction of the time dependent damped harmonic oscillator is derived.

Solution of a particle s motion in a one-dimensional infinite square potential well using path integral;

Introduction of Feynman s path integral theory into engineering physics;

A real time path integral approach is developed in order to work out a correct solution to a problem for the smaller result of the fusion probability of heavy nuclei based on the classical diffusion model at sub-barrier energies.