In the study of differential equations, a fundamental objective is to understand how a specific physical law or geometric property can be expressed in terms of rates of change. Just as a family of curves in a plane leads to an Ordinary Differential Equation (ODE), a family of surfaces in three-dimensional space leads to a Partial Differential Equation (PDE). This process involves the elimination of arbitrary parameters (constants) that define the specific members of the family, leaving behind a relationship that holds for the entire class of surfaces. On this page, we will master the systematic elimination of these constants to "construct" the governing first-order PDE.

A family of surfaces in $\mathbb{R}^3$ is often represented by a primitive equation of the form $$\begin{equation} f(x, y, z, a, b) = 0 \quad \tag{1} \end{equation}$$ where $x, y$ are independent variables, $z$ is the dependent variable (representing the surface height), and $a, b$ are arbitrary constants or parameters. Our goal is to derive a relationship involving $x, y, z$ and the partial derivatives $p = \frac{\partial z}{\partial x}$ and $q = \frac{\partial z}{\partial y}$ that is independent of $a$ and $b$.