Consider a function $f(x,y)=2x+y$. Then f(x,y) \ge f(x',y') if 2x+y \ge 2x'+y'. So $2x+y$ defines the direction that this function increases. Now I want to make sure that the rate of increase in this direction is fast enough so that it will be faster than some $\lambda >0.$ For example, $f(x,y)=0.2x+0.1y$ also increase in the same direction but the rate of increase is smaller. How can I write these conditions down in a concise way?
On the direction of fastest increase of a function of several variables
1 Answers
Given a unit vector $\vec{n}$ and a point $p=(x,y)$, one can look for the behaviour of the function $t\mapsto f(p+t\vec{n})$ when $t\to0^+$. More specifically, the rate of increase of $f$ starting from $p$ in the direction $\vec{n}$ is $ \lambda_{p,\vec{n}}=\lim\limits_{t\to0^+}\frac{f(p+t\vec{n})-f(p)}t=(\text{grad}f)(p)\cdot\vec{n}. $ When $f:(x,y)\mapsto2x+y$, it seems that you have in mind to consider the normal unit vector $\vec{n}=\frac1{\sqrt5}(2,1)$ to every line $2x+y=h$. Hence, in the general case, you would probably choose for $\vec{n}$ the gradient $(\text{grad}f)(p)$ of $f$ at the point $p$, normalized. This yields $ \lambda_p=\|(\text{grad}f)(p)\|=\sqrt{\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2}. $
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0Let me think about it. – 2012-01-13