What are the general definitions of partial differential equations (PDE)?
Defining PDEs as differential equations in which the unknown function depends on more than one independent variable.
Studying the case of two independent variables (denoted by x and y) and the unknown function denoted by z(x,y).
Representing a solution or integral of a PDE as a function z = z(x,y) that satisfies the equation by substituting the function and its partial derivatives.
Defining total differential equations or Pfaff equations as equations of the form P(x,y,z)dx + Q(x,y,z)dy + R(x,y,z)dz = 0, where P, Q, and R are differentiable functions.
Characterizing the integrability condition of a Pfaff equation as the necessary and sufficient condition that the curl of the vector field F = (P,Q,R) is zero.