# Showing that if a plane curve $L$ is always perpendicular to a fixed vector, then $L$ is a line

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!

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