80 - Pfaffian Differential Equations

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This chapter discusses Pfaffian differential equations. Another name for a Pfaffian differential equation is a total differential equation. Pfaffian differential equations are partial differential equations of the form f(x) dx = ∑Fi(x1, x2,…,xn)dxi = 0. For equations of this type, if n = 3, then a necessary and sufficient condition that f(x) dx = ∑Fi(x1, x2,.,xn)dxi = 0 be integrable is that f(x) − curl f(x) = 0. There exist a number of techniques for integrating Pfaffian equations. One way to solve Pfaffian differential equations in three dimensions is by the observation, if curl f(x) = 0, then f(x) must be the gradient of a scalar. Hence, the set of partial differential equations fi(x) = ∂v(x)/∂xi, for i = 1, …., n. The solution to f(x).dx = ∑Fi(x1, x2,….xn)dxi = 0, will then be given implicitly by v(x) = constant.