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Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a function. A level set is a set of points:

$$L(c) = \{x \in \mathbb{R}^n | f(x) = c\}$$

Two vectors $a, b \in \mathbb{R}^n$ are perpendicular, when their dot product is 0: $$a \perp b :\Leftrightarrow a \cdot b = \sum_{i=1}^n a_i b_i = 0$$

The gradient of $f$ is

$$\nabla f = \begin{pmatrix}\frac{\partial f}{\partial x_1}\\ \frac{\partial f}{\partial x_2}\\ \dots\\ \frac{\partial f}{\partial x_n}\\\end{pmatrix}$$

Question

  • Why is $\nabla f(p)$ at any given point $p \in \mathbb{R}^n$ perpendicular to the level set $L(f(p))$?
    • What does it mean anyway to be perpendicular to the level set? Does it mean the tangent of the level set in this point is perpendicular to the gradient in this point?
      • How do I get the tangent?
    • Are there any important implications of this?

Context

I found the question "why is the level curve perpendicular to the gradient" in an exam protocol for probabilistic planning.

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  • $\begingroup$ I am not sure which tags to use for this question. $\endgroup$ Jul 26, 2016 at 10:00
  • $\begingroup$ Related: math.stackexchange.com/a/603120/657156 $\endgroup$
    – Peter_Pan
    Apr 30, 2020 at 23:04
  • $\begingroup$ The gradient is the direction of steepest ascent, and the fastest way to increase the function is to go directly to the next level set, i.e. perpendicular to the current one $\endgroup$ Sep 1, 2021 at 23:34

2 Answers 2

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First of all, when dealing with more than two variables level set is a better denomination than level curve (or level surface in three dimensions.)

Now to your question. Let $x_0\in L(c)$ and let $\gamma\colon(-a,a)\to \mathbb{R}^n$ be a $C^1$ curve contained in $L(c)$ and such that $\gamma(0)=x_0$. Then $$ f(\gamma(t))=c,\quad -a<t<a. $$ Differentiating with respect to $t$ and evaluating at $t=0$ we get $$ \nabla f(x_0)\cdot\gamma'(0)=0. $$ The set of all vectors $\gamma'(0)$ for all possible curves $\gamma$ forms the tangent hyperplane to $L(c)$ at $x_0$, and $\nabla f(x_0)$ is orthogonal to all of them, that is, the gradient is orthogonal to the tangent hyperplane of the level set.

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    $\begingroup$ Hello, I am not sure if it is in poor taste to comment on a post this old, but I was wondering why it is that we can construct such curve that is also C1. It makes sense intuitively but what is the justification? $\endgroup$ Jan 12, 2018 at 18:23
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    $\begingroup$ It can always be done. A neighborhood $U$ of $x_0$ in $L(c)$ is diffeonrphic to an open set in $O\subset\Bbb R^{n-1}$. Let $\phi\colon O\to U$ be a diffeomorphism. Without loss of generality we can assume $0\in O$ and $\phi(0)=x_0$. If $\alpha\colon(-a,a)\to O$ is a $C^1$ curve in $O$ with $\phi(0)=0$, we van take $\gamma=\phi\circ\alpha$. $\endgroup$ Jan 12, 2018 at 19:28
  • $\begingroup$ For other readers – the second sentence from the comment above follows from the implicit function theorem (see here for more information). $\endgroup$
    – Rodvi
    Jun 25, 2022 at 14:12
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For your first question:

We say a vector $v \in R^n$ is perpendicular to $S \subset R^n$ at $p \in S$ if for every curve $\gamma: (-a,a) \to R^n $ s.t $Img(\gamma) \subset S$ and $\gamma(0) = p$, it holds that $\gamma'(0) \cdot v = 0$

With this definition:

Suppose that $\bigtriangledown f(p) \neq 0$

Let $\gamma:(-a,a) \to L(f(p))$ with $\gamma(0) = p$.

Notice that $\forall x \in L(f(p))$ $f(x) = f(p)$ is constant.

Thus $\forall t \in (-a,a)$ $f(\gamma(t)) = f(p)$, and we can differentiate to get $ 0 = \frac{d}{dt}f(\gamma(t)) = \bigtriangledown f(\gamma(t))\gamma'(t)$.

Setting $t = 0 $ yields $ 0 = \bigtriangledown f(\gamma(0))\gamma'(0) = \bigtriangledown f(p)\gamma'(0) $ which is what we wanted.

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