The equation $f\left(x\,y\,z\right)\=c$ can, in principle, be solved explicitly for $z\=z\left(x\,y\right)$ wherever $f_z\ne 0$.

Implicit (partial) differentiation leads to $z_x\=-f_x\/f_z$ and $z_y\=-f_y\/f_z$.

The coordinate curves ${\mathbf{R}}_1\=\left[\begin{array}{c}x\\ b\\ z\(x\,b\)\end{array}\right]$ and ${\mathbf{R}}_2\=\left[\begin{array}{c}a\\ y\\ z\(a\,y\)\end{array}\right]$ project onto the grid lines $y\=b$ and $x\=a$, respectively.

Tangents to these curves, namely, ${\mathbf{T}}_1\=\left[\begin{array}{c}1\\ 0\\ z_x\end{array}\right]$ and ${\mathbf{T}}_2\=\left[\begin{array}{c}0\\ 1\\ z_y\end{array}\right]$, are distinct vectors tangent to the surface.

Their cross product