Apologies if there is standard notation for the following that I am not aware of, but I will try to define everything that I am talking about.
Suppose we are looking at the level sets of a function $\phi(u,v)=c$ in the plane, and there is an intersection of level sets at a point.
For context, the gradient of a different function $\psi(u,v)$ points along lines of constant $\phi(u,v)$, and I am examining a saddle point of $\psi$. Its gradient will then point along two directions at the point of interest.
I want to find the slope of one of these two lines of constant $\phi$ in the plane: That is, I want to calculate $\frac{\partial v}{\partial u}$ at the saddle point for one of the lines. As a simple example, if we had $u^2-v^2=0$, then at the point (0,0) the lines $v=\pm u$ intersect, and I would want to find the slope of the v=u line. How do I go about doing this when I don't have an explicit equation for $v(u)$? Taking the total derivative $\frac{d}{du}\phi(u,v)=0$ gives an indeterminate result at the point of intersection.
Please let me know if I need to elaborate further.
To keep things simple we assume that $f(u,v)$ has a saddle point at $(0,0)$ and that $f(0,0) = 0$.
If $f$ has a saddle point it means $\frac{\partial f}{\partial u} = \frac{\partial f}{\partial v} = 0$. Which means that $f$ can be approximated by a quadratic polynomial with 0 linear part, i.e., $f(u,v) \approx au^2 + buv + cv^2$ where $a,b,c$ are corresponding second partial derivatives. Now, for an saddle point we will have $b^2 - 4ac > 0$ and the quadratic form can be represented as a product of linear forms, i.e. $au^2+buv+cv^2 = (pu+qv)(ru+sv)$, and each of the linear forms represents (locally!) a straight line, for which you can compute your slopes.