.. _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>`