- #1
Malamala
- 299
- 27
Hello! This if from a physics paper but I will write it as abstract as I can. We have a function ##f(g(a),a)## and we know that f is minimized with respect to g for any given a i.e. $$\frac{df}{dg}|_a=0$$ As this is true for any a, we have $$\frac{d}{da}\frac{df}{dg}|_a=0$$ from which we get: $$\frac{\partial}{\partial a}(\frac{df}{dg})+\frac{d^2f}{dg^2}\frac{dg}{da}|_a=0$$ And from here ##\frac{dg}{da}## can be easily extracted (which is what I need), in terms of the other derivatives (which are assumed to be known). (Here is the actual physics paper from which I simplified the equations, for reference, equations 20-23). Now I want to extent the same analysis to one more variable i.e. ##f(g(a,b),a,b)## and find an expression for ##\frac{\partial g}{\partial a}## and ##\frac{\partial g}{\partial b}## (with the same minimization assumptions) but I am kinda stuck. I tried to follow the same approach i.e. $$\frac{df}{dg}|_{a,b}=0$$ from which I would get $$\frac{\partial}{\partial a}\frac{df}{dg}|_{a,b}=0$$ and $$\frac{\partial}{\partial b}\frac{df}{dg}|_{a,b}=0$$ But I am not sure what to do from here. I guess I am writing the new equations the wrong way, but I am not sure what to do. Can someone help me? Thank you!