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| {{about| the concept from elementary [[differential calculus]]| the generalized advanced mathematical concept from [[differential topology]] and [[differential geometry]] | closed and exact differential forms}}
| | Looking at playing a new tutorial game, read the take advantage of book. Most dvds have a book you can purchase separately. You may want to consider doing this and reading it before the individual play, or even while you are playing. This way, you could certainly get the most on the market of your game run.<br><br> |
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| In [[multivariate calculus]], a [[differential (infinitesimal)|differential]] is said to be '''exact''' (or perfect), as contrasted with an [[inexact differential]], if it is of the form ''dQ'', for some differentiable [[function (mathematics)|function]] ''Q''.
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| ==Overview==
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| ===Definition===
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| We work in three dimensions, with similar definitions holding in any other number of dimensions. In three dimensions, a form of the type
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| :<math>A(x,y,z) dx + B(x,y,z) dy + C(x,y,z) dz</math>
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| is called a [[differential form]]. This form is called ''exact'' on a domain <math>D \subset \mathbb{R}^3</math> in space if there exists some scalar function <math>Q = Q(x,y,z)</math> defined on <math>D</math> such that | |
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| :<math>dQ \equiv \left ( \frac{\partial Q}{\partial x} \right )_{y,z} dx + \left ( \frac{\partial Q}{\partial y} \right )_{z,x} dy + \left ( \frac{\partial Q}{\partial z} \right )_{x,y} dz,</math> {{pad|3em}} <math>dQ = A dx + B dy + C dz</math>
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| throughout D. This is equivalent to saying that the vector field <math>(A, B, C)</math> is a [[conservative vector field]], with corresponding potential <math>Q</math>.
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| ===One dimension===
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| In one dimension, a differential form
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| :<math>A(x) \, dx</math>
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| is exact as long as <math>A</math> has an [[antiderivative]]; in this case let <math>Q</math> be the antiderivative of <math>A</math>. Otherwise, if <math>A</math> does ''not'' have an antiderivative, we cannot write <math>dQ = A(x) \, dx</math> and so the differential form is inexact.
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| ===Two and three dimensions===
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| By [[symmetry of second derivatives]], for any "nice" (non-[[Pathological (mathematics)|pathological]]) function <math>Q</math> we have
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| :<math>\frac{\partial ^2 Q}{\partial x \partial y} = \frac{\partial ^2 Q}{\partial y \partial x}</math>
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| Hence, it follows that in a [[simply-connected]] region ''R'' of the ''xy''-plane, a differential
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| :<math>A(x, y)\,dx + B(x, y)\,dy</math>
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| is an exact differential [[if and only if]] the following holds: | |
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| :<math>\left( \frac{\partial A}{\partial y} \right)_x = \left( \frac{\partial B}{\partial x} \right)_y</math>
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| For three dimensions, a differential
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| :<math>dQ = A(x, y, z) \, dx + B(x, y, z) \, dy + C(x, y, z) \, dz</math>
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| is an exact differential in a simply-connected region ''R'' of the ''xyz''-coordinate system if between the functions ''A'', ''B'' and ''C'' there exist the relations:
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| :<math>\left( \frac{\partial A}{\partial y} \right)_{x,z} \!\!\!= \left( \frac{\partial B}{\partial x} \right)_{y,z}</math> ''';''' <math>\left( \frac{\partial A}{\partial z} \right)_{x,y} \!\!\!= \left( \frac{\partial C}{\partial x} \right)_{y,z}</math> ''';''' <math>\left( \frac{\partial B}{\partial z} \right)_{x,y} \!\!\!= \left( \frac{\partial C}{\partial y} \right)_{x,z}</math>
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| ::Note: The subscripts outside the parenthesis indicate which variables are being held constant during differentiation. Due to the definition of the [[partial derivative]], these subscripts are not required, but they are included as a reminder. | |
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| These conditions are equivalent to the following one: If ''G'' is the graph of this vector valued function then for all tangent vectors ''X'',Y of the ''surface'' ''G'' then ''s''(''X'', ''Y'') = 0 with ''s'' the [[symplectic form]].
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| These conditions, which are easy to generalize, arise from the independence of the order of differentiations in the calculation of the second derivatives. So, in order for a differential ''dQ'', that is a function of four variables to be an exact differential, there are six conditions to satisfy.
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| In summary, when a differential ''dQ'' is exact:
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| *the function ''Q'' exists;
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| *<math>\int_i^f dQ=Q(f)-Q(i),</math> independent of the path followed.
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| In [[thermodynamics]], when ''dQ'' is exact, the function ''Q'' is a state function of the system. The thermodynamic functions ''[[Internal energy|U]]'', ''[[Entropy|S]]'', ''[[Enthalpy|H]]'', ''[[Helmholtz free energy|A]]'' and ''[[Gibbs free energy|G]]'' are [[state function]]s. Generally, neither [[Work (thermodynamics)|work]] nor [[heat]] is a state function. An ''exact differential'' is sometimes also called a 'total differential', or a 'full differential', or, in the study of [[differential geometry]], it is termed an [[exact form]].
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| ==Partial differential relations==
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| If three variables, <math>x</math>, <math>y</math> and <math>z</math> are bound by the condition <math>F(x,y,z) = \text{constant}</math> for some differentiable function <math>F(x,y,z)</math>, then the following [[total differential]]s exist<ref name="Cengel1998">{{cite book |last=Çengel |first=Yunus A. |authorlink= |coauthors=Boles, Michael A. |title=Thermodynamics - An Engineering Approach |origyear=1989 |edition=3rd |series=McGraw-Hill Series in [[Mechanical Engineering]] |year=1998 |publisher=McGraw-Hill |location=Boston, MA. |isbn=0-07-011927-9 |chapter=Thermodynamics Property Relations}}</ref>{{rp|667&669}}
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| :<math>d x = {\left ( \frac{\partial x}{\partial y} \right )}_z \, d y + {\left ( \frac{\partial x}{\partial z} \right )}_y \,dz</math>
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| :<math>d z = {\left ( \frac{\partial z}{\partial x} \right )}_y \, d x + {\left ( \frac{\partial z}{\partial y} \right )}_x \,dy.</math>
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| Substituting the first equation into the second and rearranging, we obtain<ref name="Cengel1998"/>{{rp|669}}
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| :<math>d z = {\left ( \frac{\partial z}{\partial x} \right )}_y \left [ {\left ( \frac{\partial x}{\partial y} \right )}_z d y + {\left ( \frac{\partial x}{\partial z} \right )}_y dz \right ] + {\left ( \frac{\partial z}{\partial y} \right )}_x dy,</math>
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| :<math>d z = \left [ {\left ( \frac{\partial z}{\partial x} \right )}_y {\left ( \frac{\partial x}{\partial y} \right )}_z + {\left ( \frac{\partial z}{\partial y} \right )}_x \right ] d y + {\left ( \frac{\partial z}{\partial x} \right )}_y {\left ( \frac{\partial x}{\partial z} \right )}_y dz,</math>
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| :<math>\left [ 1 - {\left ( \frac{\partial z}{\partial x} \right )}_y {\left ( \frac{\partial x}{\partial z} \right )}_y \right ] dz = \left [ {\left ( \frac{\partial z}{\partial x} \right )}_y {\left ( \frac{\partial x}{\partial y} \right )}_z + {\left ( \frac{\partial z}{\partial y} \right )}_x \right ] d y.</math> | |
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| Since <math>y</math> and <math>z</math> are independent variables, <math>d y</math> and <math>d z</math> may be chosen without restriction. For this last equation to hold in general, the bracketed terms must be equal to zero.<ref name="Cengel1998"/>{{rp|669}}
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| ===Reciprocity relation===
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| Setting the first term in brackets equal to zero yields<ref name="Cengel1998"/>{{rp|60฿฿฿70}}
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| :<math>{\left ( \frac{\partial z}{\partial x} \right )}_y {\left ( \frac{\partial x}{\partial z} \right )}_y = 1.</math>
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| A slight rearrangement gives a reciprocity relation,<ref name="Cengel1998"/>{{rp|670}}
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| :<math>{\left ( \frac{\partial z}{\partial x} \right )}_y = \frac{1}{{\left ( \frac{\partial x}{\partial z} \right )}_y}.</math>
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| There are two more [[permutations]] of the foregoing derivation that give a total of three reciprocity relations between <math>x</math>, <math>y</math> and <math>z</math>. [[Inverse functions and differentiation|Reciprocity relations]] show that the inverse of a partial derivative is equal to its reciprocal.
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| ===Cyclic relation===
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| The cyclic relation is also known as the cyclic rule or the [[Triple product rule]]. Setting the second term in brackets equal to zero yields<ref name="Cengel1998"/>{{rp|670}}
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| :<math>{\left ( \frac{\partial z}{\partial x} \right )}_y {\left ( \frac{\partial x}{\partial y} \right )}_z = - {\left ( \frac{\partial z}{\partial y} \right )}_x.</math>
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| Using a reciprocity relation for <math>\tfrac{\partial z}{\partial y}</math> on this equation and reordering gives a cyclic relation (the [[triple product rule]]),<ref name="Cengel1998"/>{{rp|670}}
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| :<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>
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| If, ''instead'', a reciprocity relation for <math>\tfrac{\partial x}{\partial y}</math> is used with subsequent rearrangement, a [[Implicit function#Formula for two variables|standard form for implicit differentiation]] is obtained:
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| :<math>{\left ( \frac{\partial y}{\partial x} \right )}_z = - \frac { {\left ( \frac{\partial z}{\partial x} \right )}_y }{ {\left ( \frac{\partial z}{\partial y} \right )}_x }.</math>
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| == Some useful equations derived from exact differentials in two dimensions ==
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| (See also [[Bridgman's thermodynamic equations]] for the use of exact differentials in the theory of [[thermodynamic equations]])
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| Suppose we have five state functions <math>z,x,y,u</math>, and <math>v</math>. Suppose that the state space is two dimensional and any of the five quantities are exact differentials. Then by the [[chain rule]]
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| <math>(1)~~~~~
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| dz =
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| \left(\frac{\partial z}{\partial x}\right)_y dx+
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| \left(\frac{\partial z}{\partial y}\right)_x dy
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| =
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| \left(\frac{\partial z}{\partial u}\right)_v du
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| +\left(\frac{\partial z}{\partial v}\right)_u dv
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| </math>
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| but also by the chain rule:
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| <math>(2)~~~~~
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| dx =
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| \left(\frac{\partial x}{\partial u}\right)_v du
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| +\left(\frac{\partial x}{\partial v}\right)_u dv
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| </math>
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| and
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| <math>(3)~~~~~
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| dy=
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| \left(\frac{\partial y}{\partial u}\right)_v du
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| +\left(\frac{\partial y}{\partial v}\right)_u dv
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| </math>
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| so that:
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| <math>(4)~~~~~
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| dz =
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| \left[
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| \left(\frac{\partial z}{\partial x}\right)_y
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| \left(\frac{\partial x}{\partial u}\right)_v
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| +
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| \left(\frac{\partial z}{\partial y}\right)_x
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| \left(\frac{\partial y}{\partial u}\right)_v
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| \right]du
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| </math>
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| :::<math>+
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| \left[
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| \left(\frac{\partial z}{\partial x}\right)_y
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| \left(\frac{\partial x}{\partial v}\right)_u
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| +
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| \left(\frac{\partial z}{\partial y}\right)_x
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| \left(\frac{\partial y}{\partial v}\right)_u
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| \right]dv
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| </math>
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| which implies that:
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| <math>(5)~~~~~
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| \left(\frac{\partial z}{\partial u}\right)_v
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| =
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| \left(\frac{\partial z}{\partial x}\right)_y
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| \left(\frac{\partial x}{\partial u}\right)_v
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| +
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| \left(\frac{\partial z}{\partial y}\right)_x
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| \left(\frac{\partial y}{\partial u}\right)_v
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| </math>
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| Letting <math>v=y</math> gives:
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| <math>(6)~~~~~
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| \left(\frac{\partial z}{\partial u}\right)_y
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| =
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| \left(\frac{\partial z}{\partial x}\right)_y
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| \left(\frac{\partial x}{\partial u}\right)_y
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| </math>
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| Letting <math>u=y</math> gives:
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| <math>(7)~~~~~
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| \left(\frac{\partial z}{\partial y}\right)_v
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| =
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| \left(\frac{\partial z}{\partial y}\right)_x
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| +
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| \left(\frac{\partial z}{\partial x}\right)_y
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| \left(\frac{\partial x}{\partial y}\right)_v
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| </math>
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| Letting <math>u=y</math>, <math>v=z</math> gives:
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| <math>(8)~~~~~
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| \left(\frac{\partial z}{\partial y}\right)_x
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| = -
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| \left(\frac{\partial z}{\partial x}\right)_y
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| \left(\frac{\partial x}{\partial y}\right)_z
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| </math>
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| using (<math>\partial a/\partial b)_c = 1/(\partial
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| b/\partial a)_c</math> gives the [[triple product rule]]:
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| <math>(9)~~~~~
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| \left(\frac{\partial z}{\partial x}\right)_y
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| \left(\frac{\partial x}{\partial y}\right)_z
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| \left(\frac{\partial y}{\partial z}\right)_x
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| =-1
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| </math>
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| == See also ==
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| *[[Closed and exact differential forms]] for a higher-level treatment
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| *[[Differential (mathematics)]]
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| *[[Inexact differential]]
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| *[[Integrating factor]] for solving non-exact differential equations by making them exact
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| *[[Exact differential equation]]
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| == References ==
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| <references/>
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| *Perrot, P. (1998). ''A to Z of Thermodynamics.'' New York: Oxford University Press.
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| *Zill, D. (1993). ''A First Course in Differential Equations, 5th Ed.'' Boston: PWS-Kent Publishing Company.
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| ==External links==
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| *[http://mathworld.wolfram.com/InexactDifferential.html Inexact Differential] – from Wolfram MathWorld
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| *[http://www.chem.arizona.edu/~salzmanr/480a/480ants/e&idiff/e&idiff.html Exact and Inexact Differentials] – University of Arizona
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| *[http://farside.ph.utexas.edu/teaching/sm1/lectures/node36.html Exact and Inexact Differentials] – University of Texas
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| *[http://mathworld.wolfram.com/ExactDifferential.html Exact Differential] – from Wolfram MathWorld
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| {{DEFAULTSORT:Exact Differential}}
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| [[Category:Thermodynamics]]
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| [[Category:Multivariable calculus]]
| |
Looking at playing a new tutorial game, read the take advantage of book. Most dvds have a book you can purchase separately. You may want to consider doing this and reading it before the individual play, or even while you are playing. This way, you could certainly get the most on the market of your game run.
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Judgements There are a involving Apple fans who play the above game all across the globe. This generation has donrrrt been the JRPG's best; in fact it's for ages been unanimously its worst. Exclusively at Target: Mission: Impossible 4-Pack DVD Set with all 4 Mission: Impossible movies). Though it is a special day of grand gifts and gestures, one Valentines Day is likely to blend into another much too easily. clash of clans is regarded as the the quickest rising video games as of late.
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