Elimination of Constants by Partial Differentiation

Theorem
Let $x_1, x_2, \dotsc, x_m$ be independent variables.

Let $c_1, c_2, \dotsc, c_n$ be arbitrary constants.

Let this equation:
 * $(1): \quad \map f {x_1, x_2, \dotsc, x_m, z, c_1, c_2, \dotsc, c_n} = 0$

define a dependent variable $z$ via the implicit function $f$.

Then it may be possible to eliminate the constants by successive partial differentiation of $(1)$.

Proof
We differentiate $(1)$ partially each of $x_j$ for $1 \le j \le m$:


 * $(2): \quad \dfrac {\partial f} {\partial x_j} + \dfrac {\partial f} {\partial z} \cdot \dfrac {\partial z} {\partial x_j} = 0$

Suppose $m \ge n$.

Then there exist sufficient equations of the form of $2$ for the constants $c_1, c_2, \dotsc, c_n$ to be eliminated.

Otherwise suppose $m > n$.

Then we differentiate $(2)$ partially for each $j$ each of $x_j$ for $1 \le j \le m$, to obtain:

and continue the operation until enough equations have been obtained so that the remaining constants can be eliminated.