Finding function given ratio of derivatives w.r. to different variables

Question

Is there any way to find a function $f(a,b)$ up to some factors and a constant, given the ratio $$\frac{\partial f / \partial a}{\partial f/ \partial b}$$? If not, is there any other useful information about $f$ one can get from this ratio or a good numerical method to approximate $f$? Thanks for helping me out.

Answer 1

Suppose we know $$ \frac{f_a}{f_b}=r(a,b). $$ Then at a point $(a,b)$, the directional derivative of $f$ in the direction $[1,-r(a,b)]$ is $0$. This implies that $f$ is constant along the solution curves to the ODE $$ \frac{db}{da}=-r(a,b). $$

In your example $$ \frac{f_a}{f_b}=\frac{2b}{a}, $$ the ODE we get is $$ \frac{db}{da}=-\frac{2b}{a}, $$ which is separable. The solution curves are of the form $a^2b=C$, where $C$ is a constant. If $g$ is any function of one variable, then $g(a^2b)$ will be constant along the curves $a^2b=C$, so will give a solution to your original equation.

Answer 2

Not sure this is what you were looking for but $f(a,b)=\lambda a+b$ has $$\frac{\frac{\partial f}{\partial a}}{\frac{\partial f}{\partial b}}=\frac \lambda 1=\lambda$$