.. _exactdiff:

Complement 1: Exact Differential
++++++++++++++++++++++++++++++++

  
Summary
-------
.. attention::

   One difficulty in thermodynamics is to establish a formal link
   between *what makes sense* and the mathematics used to describe
   it. At a central point of the first and second laws of
   thermodynamics is the concept of **function of states**. 

   A function of state :math:`f(x)` is a function that only depends on
   the state the system is in, it does not depend on *how it got
   there*.

   Mathematically, such a function must be an exact differential such that

   .. math::
      
         \int_i^f \mathrm{d} f = f(x_f)-f(x_i)

   or, equivalently: 
	 
   .. math::

      \oint \mathrm{d} f = 0

   Conversely, a function that is **not** a function of state is one
   that depends *on how it got there*. This is an important distinction
   as the path taken by a system will necessarily yield different
   values for such a function. Mathematically, this translates into the
   fact that the function is an **inexact** differential. We introduce
   a notation that reminds us that the equations above do not apply,
   namely, for a function :math:`g(x)`, which is not a function of
   state, we have

   
   .. math::
      
      \int_i^f \dbar(g) \neq  g(x_f)-g(x_i)

   and of course

   .. math::
      
      \oint \dbar(g) \neq  0

	 
Partial differential
--------------------

Consider a function :math:`x=x(y,z)`, then we have:

.. math::
   
   \mathrm{d} x=\left(\frac{\partial x}{\partial y}\right)_{z} \mathrm{~d} y+\left(\frac{\partial x}{\partial z}\right)_{y} \mathrm{~d} z

Similarly, for :math:`z=z(x,y)`, we have:

.. math::
   
   \mathrm{d} z=\left(\frac{\partial z}{\partial x}\right)_{y} \mathrm{~d} x+\left(\frac{\partial z}{\partial y}\right)_{x} \mathrm{~d} y

It follows, from those two equations that:

.. math::

   \mathrm{d} x=\left(\frac{\partial x}{\partial z}\right)_{y}\left(\frac{\partial z}{\partial x}\right)_{y} \mathrm{~d} x+\left[\left(\frac{\partial x}{\partial y}\right)_{z}+\left(\frac{\partial x}{\partial z}\right)_{y}\left(\frac{\partial z}{\partial y}\right)_{x}\right] \mathrm{d} y

It follows two important theorems:

:index:`Reciprocal theorem`:
   .. math::
      \left(\frac{\partial x}{\partial z}\right)_{y}=\frac{1}{\left(\frac{\partial z}{\partial x}\right)_{y}}

index:`Reciprocity theorem`:
   .. math::
      \left(\frac{\partial x}{\partial y}\right)_{z}\left(\frac{\partial y}{\partial z}\right)_{x}\left(\frac{\partial z}{\partial x}\right)_{y}=-1

Corollary
   By combining the two theorems, we find a very handy relationship:

   .. math:: 
      \left(\frac{\partial x}{\partial y}\right)_{z}=-\left(\frac{\partial x}{\partial z}\right)_{y}\left(\frac{\partial z}{\partial y}\right)_{x}


Exact differential
------------------

Consider the function :math:`f(x,y)`. This function will be an exact differential provided that

.. math::
   
   \left(\frac{\partial^{2} f}{\partial x \partial y}\right)=\left(\frac{\partial^{2} f}{\partial y \partial x}\right)

This relation can be proven using Stokes theorem.

Copy of Slides
~~~~~~~~~~~~~~

The slides for this complement are available in pdf format here:  :download:`pdf <_pdfs/slides/exactdif.pdf>`