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The gradient of a differentiable function $f:S\to \mathbb R$ is a differentiable map grad $f:S\to \mathbb R^3$ which assigns to each point $p\in S$ a vector grad $f(p)\in T_p(S)\subset \mathbb R^3$ such that $$\langle \operatorname{grad}{f(p)}, v\rangle_p=df_p(v) \ \text{ for all } v\in T_p(S).$$

Show that

If grad $f\neq0$ at all points of the level curve $C=\{q\in S; f(q)=\text{ constant}\}$, then $C$ is a regular curve on S and grad $f$ is normal to $C$ at all points of $C$.

Here is my working, can anyone please check it?

  1. If grad $f\neq0$ and $<\text{grad }f, v>=0$ for all non-zero $v\in C$, then grad $f$ is normal to $C$ at all points of $C$. Am I correct?

  2. Let $\alpha:(\epsilon, \epsilon)\to C$ be a differentiable curve such that $\alpha(0)=p$, $\alpha'(0)=v$ for $p\in C$. Then $f\circ\alpha:(\epsilon, \epsilon)\to\mathbb{R}$. Then $df_p(v)=\frac{df}{d\alpha}\frac{d\alpha}{dt}\bigg|_{t=0}$. Here is the part that confuses me, what is $\frac{df}{d\alpha}$? since $f$ is a constant, so is it zero? that means $df_p(v)=0$, so how can it be a regular curve?

Could anyone please give some help on how to fix my mistakes?

Many thanks in advance!

user71346
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  • $\langle \text{grad} f, v\rangle$ for $v\in C$ don't quite make sense as $C$ is not a subset of tangent vectors. $C$ is in $S$. 2. I think you should tell us what is $\frac{df}{d\alpha}$. You said that this is your working. Or if not, at least tell us where you find that.
    • –  Jan 08 '16 at 10:45
  • Have you made a sketch? 2. Have you seen the implicit function theorem? 3. When you write "$C$ is a regular curve in $S$", does that mean "$C$ is locally the image of a regular curve in $S$", i.e., "for every point $p$ of $C$, there exists an open neighborhood $U$ of $p$ in $S$ and a regular curve $\gamma:(-1, 1) \to S$ such that $\gamma(0) = p$, $\gamma'(0) \neq 0$, and $\gamma(-1, 1) = U \cap C$"?
    • – Andrew D. Hwang Jan 08 '16 at 12:56
  • @JohnMa $\frac{df}{d\alpha}\frac{d\alpha}{dt}$ is a chain rule used in DoCarmo's definition of $df_p{v}$. And in DoCarmo's textbook, there is only a definition of regular curve in terms of a parametrised curve, I find it hard to relate to surfaces and to $df_p{v}$. – user71346 Jan 15 '16 at 11:34
  • @AndrewD.Hwang I haven't sketched because I wasn't sure how to sketch. Yes I have seen the implicit function theorem and it tells us that there exists a function of one variables whose graph is a function of the original variable. But I'm not sure how to apply it in this problem? – user71346 Jan 15 '16 at 11:46