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I am working on the problem

Suppose $f : \mathbb{R}^2 \to \mathbb{R}$ is a differentiable function whose gradient is nowhere zero and satisfies $\frac {\partial f}{\partial x} = 2 \frac {\partial f}{\partial y}$ everywhere. Find the level curves of $f$.

Source: Multivariable Mathematics by Ted Shifrin, page 120.

I think I've mostly solved this, but I have several doubts about my solution. Could you please enlighten me about the dubious steps? Thank you!

Partial Solution

We see $$\nabla f = [ \frac {\partial f}{\partial x} \ \ \ \frac {\partial f}{\partial y} ]^T = \frac {\partial f}{\partial y} [2 \ \ \ 1]^T$$ and $\frac {\partial f}{\partial y}$ is never $0$. Let $L$ be the level curve $L = \{(x, y)|f(x, y) = c \},$ and write it in parametric form $[x(t), y(t)]^T.$ Since the gradient is perpendicular to the level curve, we must have $$\forall t: \ \ 2x'(t) + y'(t) = 0$$ so $y(t) = - \frac 12 x(t) + k$ for some constant $k$. Thus, we see that $L$ must be a $\textit{subset}$ of a line.

My questions are:

  1. Can we prove that $L$ is indeed an entire, infinite line?

  2. How can we justify that we can write the set $L$ as the image of $[x(t) \ \ \ y(t)]$ for some functions $x, y$?

Thank you!

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  • $\begingroup$ Proceed to find $f(x,y)$, which is defined on all of $\mathbb{R}^2$, see the method of characteristics. $\endgroup$
    – Conifold
    Oct 5, 2021 at 17:39
  • $\begingroup$ But any such line is globally parametrized and $f$ is constant on the entire line. $\endgroup$ Oct 5, 2021 at 17:57
  • $\begingroup$ @TedShifrin Wow it's great to see the author himself :) You are responding to Conifold's comment and not to the question, right? $\endgroup$
    – user56202
    Oct 5, 2021 at 19:34
  • $\begingroup$ Nope, I was responding to you :P $\endgroup$ Oct 5, 2021 at 19:38
  • $\begingroup$ @TedShifrin Thanks, I will have to think more about this. $\endgroup$
    – user56202
    Oct 5, 2021 at 19:46

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