one problem on multivariable claculus

Question

Suppose $\phi(\bar{x}(t))$ be a function which takes vectors (parameterized by $t$) as argument. Now take $c$ be a minimum point of the function $\phi$. consider a curve $\gamma(t)$ which passes through the minimum point. consider the $t$ where $\gamma(t)$ lies near $c$. Now, if $\frac{d\phi}{dt}$ > 0 (or, < 0) at that $t$, what can we say about the curve $\gamma$ is it moving toward $c$ or, moving away from $c$.

Answer 1

I am not sure whether I got your question is right! Kindly correct me if I am wrong.

Now $\phi$ is a scalar function I assume, and also I am assuming it is the absolute minimum you are talking about (not saddle point).

I believe the answer lies in the gradient function,

$\vec {\nabla \phi} = < \frac{\partial \phi}{\partial x}, \frac{\partial \phi}{\partial y}, \frac{\partial \phi}{\partial z}>$

This tells us the direction moving away from or to the minimum point.

Now $\frac{d\phi}{dt} = \frac{\partial \phi}{\partial x}x'(t) + \frac{\partial \phi}{\partial y}y'(t) +\frac{\partial \phi}{\partial z}z'(t)$. Now lets take a small element along curve $\gamma (t)$. Which would be $\vec {dr} = <x'(t), y'(t), z'(t)>dt$. And now we have to take a dot product with the gradient.

$\vec {\nabla \phi}.\vec {dr} = \frac{d\phi}{dt} dt$. Now clearly if $\frac{d\phi}{dt} > 0$, then the curve $\gamma(t)$ is moving away from the minimum point as the direction of the curve in increasing $t$ is towards the gradient direction. Whereas if $\frac{d\phi}{dt} < 0$, then increasing $t$ would move it towards the minimum point as now the direction of the curve in increasing $t$ is in opposite direction to the gradient.

Note: I have tried to provide an answer but still there can be mistakes, would greatly appreciate possible corrections.