Name and uniqueness of a concept

Question

Consider a the gradient of a function. Starting from a point in the domain, I can follow a trajectory such that its tangent is parallel to the gradient of the function in every point.

Is this trajectory unique? What is its name?

Answer 1

Consider the function $f(x, y) = x^2 + y^2$. The gradient is $\langle 2x, 2y\rangle$, so if you depart from the origin at any angle and continue straight, you will be parallel to the gradient at any point. So the trajectory isn't unique; you'll have to choose an initial velocity.

For any function $f$, I believe we can identify one of these trajectories as a path $r(t) :: \mathbb{R} \rightarrow \mathbb{R}^n$ that solves the differential equation:

$$r'(t) = \nabla f$$

If we write $r(t) = \langle x(t), y(t) \rangle$, then this means we have a system of differential equations:

$$ \begin{aligned} x'(t) &= \frac{\partial}{\partial x} f\\ y'(t) &= \frac{\partial}{\partial y} f\\ \end{aligned} $$

and we can find $\frac{\partial}{\partial x}f$ in terms of $t$ by using the Chain Rule. I'd appreciate if someone could do that out.

As far as a name -- I don't know exactly! Certainly seems related to gradient descent/ascent.