Abstract: The commonly used mathematical tools, such as derivative and total differentiation of function of many variables, Jacobian determinant, Legendre transformation and derivative of complex functions are listed, respectively. Using the mathematical tools, six proving method of thermodynamic relation is summed in terms of the basic law of thermodynamics and fully analysing relation characteristic. The main points and examples for every kind of methods are given.

Key words: thermodynamic, Legendre transformation, Jacobian determinant