Editing Exact differential (section)

===Two and three dimensions===
By [[symmetry of second derivatives]], for any "well-behaved" (non-[[Pathological (mathematics)|pathological]]) function <math>Q</math>, we have

: <math>\frac{\partial ^2 Q}{\partial x \, \partial y} = \frac{\partial ^2 Q}{\partial y \, \partial x}.</math>

Hence, in a [[simply-connected]] region ''R'' of the ''xy''-plane, where <math>x,y</math> are independent,<ref name=":0">If the pair of independent variables <math>(x,y)</math> is a (locally reversible) function of dependent variables <math>(u,v)</math>, all that is needed for the following theorem to hold, is to replace the partial derivatives with respect to <math>x</math> or to <math>y</math>, by the partial derivatives with respect to <math>u</math> and to <math>v</math> involving their [[Jacobian]] components. That is: <math>A(u, v)du + B(u, v)dv,</math> is an exact differential, if and only if: <math>\frac{\partial A}{\partial u}\frac{\partial u}{\partial y} + \frac{\partial A}{\partial v}\frac{\partial v}{\partial y} = \frac{\partial B}{\partial u}\frac{\partial u}{\partial x} + \frac{\partial B}{\partial v}\frac{\partial v}{\partial x}.</math></ref> a differential form

:<math>A(x, y)\,dx + B(x, y)\,dy</math>

is an exact differential if and only if the equation

:<math>\left( \frac{\partial A}{\partial y} \right)_x = \left( \frac{\partial B}{\partial x} \right)_y</math>

holds. If it is an exact differential so <math>A=\frac{\partial Q}{\partial x}</math> and <math>B=\frac{\partial Q}{\partial y}</math>, then <math>Q</math> is a differentiable (smoothly continuous) function along <math>x</math> and <math>y</math>, so <math>\left( \frac{\partial A}{\partial y} \right)_x = \frac{\partial ^2 Q}{\partial y \partial x} = \frac{\partial ^2 Q}{\partial x \partial y} = \left( \frac{\partial B}{\partial x} \right)_y</math>. If <math>\left( \frac{\partial A}{\partial y} \right)_x = \left( \frac{\partial B}{\partial x} \right)_y</math> holds, then <math>A</math> and <math>B</math> are differentiable (again, smoothly continuous) functions along <math>y</math> and <math>x</math> respectively, and <math>\left( \frac{\partial A}{\partial y} \right)_x = \frac{\partial ^2 Q}{\partial y \partial x} = \frac{\partial ^2 Q}{\partial x \partial y} = \left( \frac{\partial B}{\partial x} \right)_y</math> is only the case.

For three dimensions, in a simply-connected region ''R'' of the ''xyz''-coordinate system, by a similar reason, a differential

:<math>dQ = A(x, y, z) \, dx + B(x, y, z) \, dy + C(x, y, z) \, dz</math>

is an exact differential if and only if between the functions ''A'', ''B'' and ''C'' there exist the relations

:<math>\left( \frac{\partial A}{\partial y} \right)_{x,z} \!\!\!= \left( \frac{\partial B}{\partial x} \right)_{y,z}</math>''';'''&nbsp;<math>\left( \frac{\partial A}{\partial z} \right)_{x,y} \!\!\!= \left( \frac{\partial C}{\partial x} \right)_{y,z}</math>''';'''&nbsp;<math>\left( \frac{\partial B}{\partial z} \right)_{x,y} \!\!\!= \left( \frac{\partial C}{\partial y} \right)_{x,z}.</math>

These conditions are equivalent to the following sentence: If ''G'' is the graph of this vector valued function then for all tangent vectors ''X'',''Y'' of the ''surface'' ''G'' then ''s''(''X'',&nbsp;''Y'')&nbsp;=&nbsp;0 with ''s'' the [[symplectic form]].

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 (the [[combination]] <math>C(4,2)=6</math>) to satisfy.