Exact differential

The exact differential of a multivariate function f(x,y):R2→Rf(x,y):\mathbb{R}^{2}\to \mathbb{R} is defined as the differential form

df=∂f∂xdx+∂f∂ydydf=\frac{ \partial f }{ \partial x } dx+\frac{ \partial f }{ \partial y } dy

where the arguments xx and yy are independent of each other. It is a specific form of the Differential for a Scalar field. This is a specific case of the general differential form

df=g(x,y)dx+h(x,y)dydf=g(x,y)dx+h(x,y)dy

where g=∂f∂xg=\frac{ \partial f }{ \partial x } and h=∂f∂yh=\frac{ \partial f }{ \partial y }. The definition can be extended to an arbitrary NN-dimensional function f(x1,…,xn):RN→Rf(x_{1},\ldots,x_{n}):\mathbb{R}^{N}\to \mathbb{R} as

df=∑i=1N∂f∂xidxidf=\sum_{i=1}^{N} \frac{ \partial f }{ \partial x_{i} }dx_{i}

dfdf is exact if and only if ff is a multivariate function whose arguments are independent of each other, that is, changing one has no effect on the others.

The benefit of an exact differential is that any integral over it is independent of the path chosen, such that for any two paths γ1\gamma_{1} and γ2\gamma_{2} defined in ff's domain we have

∫γ1df=∫γ2df\int_{\gamma_{1}}df=\int_{\gamma_{2}}df

In thermodynamics, a function that has an exact differential and determines the state of a physical system is called an equation of state.