In order for the above form to be nonholonomic, it is also required that the left hand side neither be a total differential nor be able to be converted into one, perhaps via an integrating factor.
22.
They will also find themselves unable to understand integration by parts, they will have difficulty solving differential equations using integrating factors, they will be surprised by the form of the multivariable chain rule, and so forth.
23.
The basic idea is to find some function M ( x ), called the " integrating factor, " which we can multiply through our linear differential equation shown above, our integrating factor is chosen to be
24.
In order to derive this, let M ( x ) be the integrating factor of a first order, linear differential equation such that multiplication by M ( x ) transforms a partial derivative into a total derivative, then:
25.
For first-order inhomogeneous linear differential equations it is usually possible to find solutions via integrating factors or undetermined coefficients with considerably less effort, although those methods leverage heuristics that involve guessing and don't work for all inhomogeneous linear differential equations.
26.
For example, thermodynamic work is a holonomic process function since the integrating factor \ lambda = 1 / p ( where p is pressure ) will yield exact differential of the volume state function dV = \ delta W / p.
27.
It is commonly used to solve ordinary differential equations, but is also used within multivariable calculus when multiplying through by an integrating factor allows an inexact differential to be made into an exact differential ( which can then be integrated to give a scalar field ).
28.
The second law of thermodynamics as stated by Carath�odory essentially amounts to the statement that heat is a holonomic process function since the integrating factor \ lambda = 1 / T ( where T is temperature ) will yield the exact differential of an entropy state function dS = \ delta Q / T.
