en>Edward: link mathematical optimization using Find link

2013-11-20T10:12:07Z

link mathematical optimization using Find link

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			The '''triple product rule''', known variously as the '''cyclic chain rule''', '''cyclic relation''', '''cyclical rule''' or '''Euler's chain rule''', is a formula which relates [[partial derivative]]s of three interdependent variables. The rule finds application in [[thermodynamics]], where frequently three variables can be related by a function of the form ''f''(''x'', ''y'', ''z'') = 0, so each variable is given as an implicit function of the other two variables. For example, an [[equation of state]] for a [[fluid]] relates [[temperature]], [[pressure]], and [[volume]] in this manner. The triple product rule for such interrelated variables ''x'', ''y'', and ''z'' comes from using a [[Inverse functions and differentiation\|reciprocity relation]] on the [[Implicit function#Formula for two variables\|result of the implicit function theorem in two variables]] and is given by

			:<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.</math>
			::Note: The third variable is considered to be an implicit function of the other two.

			Here the subscripts indicate which variables are held constant when the partial derivative is taken. That is, to explicitly compute the partial derivative of ''x'' with respect to ''y'' with ''z'' held constant, one would write ''x'' as a function of ''y'' and ''z'' and take the partial derivative of this function with respect to ''y'' only.

			The advantage of the triple product rule is that by rearranging terms, one can derive a number of substitution identities which allow one to replace partial derivatives which are difficult to analytically evaluate, experimentally measure, or integrate with quotients of partial derivatives which are easier to work with. For example,

			:<math>\left(\frac{\partial x}{\partial y}\right)_z = - \frac{\left(\frac{\partial z}{\partial y}\right)_x}{\left(\frac{\partial z}{\partial x}\right)_y}</math>

			Various other forms of the rule are present in the literature; these can be derived by permuting the variables {''x'', ''y'', ''z''}.

			== Derivation ==

			An informal derivation follows. Suppose that ''f''(''x'', ''y'', ''z'') = 0. Write ''z'' as a function of ''x'' and ''y''. Thus the [[total derivative]] ''dz'' is

			:<math>dz = \left(\frac{\partial z}{\partial x}\right)_y dx + \left(\frac{\partial z}{\partial y}\right)_x dy</math>

			Suppose that we move along a curve with ''dz'' = 0, where the curve is parameterized by ''x''. Thus ''y'' can be written in terms of ''x'', so on this curve

			:<math>dy = \left(\frac{\partial y}{\partial x}\right)_z dx</math>

			Therefore the equation for ''dz'' = 0 becomes

			:<math>0 = \left(\frac{\partial z}{\partial x}\right)_y \, dx + \left(\frac{\partial z}{\partial y}\right)_x \left(\frac{\partial y}{\partial x}\right)_z \, dx</math>

			Since this must be true for all ''dx'', rearranging terms gives

			:<math>\left(\frac{\partial z}{\partial x}\right)_y = -\left(\frac{\partial z}{\partial y}\right)_x \left(\frac{\partial y}{\partial x}\right)_z</math>

			Dividing by the derivatives on the right hand side gives the triple product rule

			:<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</math>

			Note that this proof makes many implicit assumptions regarding the existence of partial derivatives, the existence of the [[exact differential]] ''dz'', the ability to construct a curve in some [[Neighbourhood_(mathematics)\|neighborhood]] with ''dz'' = 0, and the nonzero value of partial derivatives and their reciprocals. A formal proof based on [[mathematical analysis]] would eliminate these potential ambiguities.

			== See also ==
			* [[Exact differential]] (has another derivation of the triple product rule)
			* [[Total derivative]]
			* [[Triple product]] for vectors and scalars.

			== References ==
			* Elliott, JR, and Lira, CT. ''Introductory Chemical Engineering Thermodynamics'', 1st Ed., Prentice Hall PTR, 1999. p. 184.
			* Carter, Ashley H. ''Classical and Statistical Thermodynamics'', Prentice Hall, 2001, p. 392.

			[[Category:Thermodynamics]]
			[[Category:Multivariable calculus]]

en>ChrisGualtieri: /* Algorithmic considerations */Typo fixing, typos fixed: ie → i.e. using AWB

2012-07-19T13:48:06Z

Algorithmic considerations: Typo fixing, typos fixed: ie → i.e. using AWB

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en>Edward: link mathematical optimization using Find link

en>ChrisGualtieri: /* Algorithmic considerations */Typo fixing, typos fixed: ie → i.e. using AWB