.. _exactdiff:

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

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

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

   A function of state :math:`f(x)` depends only 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 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 to the
   fact that the function is an **inexact** differential. To remind us
   that the equations above do not apply, we introduce
   a different notation. For a function :math:`g(x)` that is not a function of
   state, we write:

   .. 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

From this we derive 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 is 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>`