Exact differential

In multivariate calculus, a differential is said to be exact or perfect, as contrasted with an inexact differential, if it is of the form ${\textstyle dQ=A(x,y,z)\,dx+B(x,y,z)\,dy+C(x,y,z)\,dz}$ for some differentiable function Q and ${\textstyle \int dQ}$ is path independent.

We work in three dimensions, with similar definitions holding in any other number of dimensions. In three dimensions, a form of the type

is called a differential form. This form is called exact on a domain ${\displaystyle D\subset \mathbb {R} ^{3}}$ in space if there exists some scalar function ${\displaystyle Q=Q(x,y,z)}$ defined on ${\displaystyle D}$ such that

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