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This question might sound stupid but I want to confirm an answer from it.

I saw somewhere online that it means that when the directional derivative of function $f$ along the none zero vector $v$ at certain point is equal to $0$, it means that the function $f$ is constant in that direction. But what does "constant in direction" mean? can anyone give me an example of it such as $f(x,y)$ to explain this?

Thanks!

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    $\begingroup$ I think it simply means that the value of the function does not change as you move along the direction given by that vector. I sounds redundant, I guess. Hope this helps $\endgroup$ – Vishesh Sep 13 '13 at 12:38
  • $\begingroup$ As an afterthought, this happens when you look at tangent vectors to level curves of the given function. Also if you know $ D_{v}f = \nabla f . v $, where $v$ is your tangent vector. The gradient is always normal to a level curve of a function. $\endgroup$ – Vishesh Sep 13 '13 at 12:43
  • $\begingroup$ But if the value long curve of that direction doesn't change, doesn't it imply that the curve is a constant? But hardly can I imagine a concrete function like this. $\endgroup$ – Cancan Sep 13 '13 at 12:54
  • $\begingroup$ Other better answers have already been given. Anyway for the sake of completion, just take a look at $ f(x,y)= 2e^{x}+3e^{y}$ at $(0,0)$ Then use the vector $ (-3,2) $. Cheers $\endgroup$ – Vishesh Sep 13 '13 at 13:05
  • $\begingroup$ What do you mean by saying a curve is a constant??? $\endgroup$ – Vishesh Sep 13 '13 at 13:06
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If $D_v f(x_0)=0$, then $f$ is constant to first order in the direction $v$ - that is, if you consider the values of $f$ along the line in the direction $v$, you find that there is no linear-order term:

$$ f(x_0+tv) = f(x_0) + t D_vf(x_0) + o(t) = f(x_0) + o(t). $$

(Here $o(t)$ is some function such that $o(t)/t \to 0$ as $t \to 0$, which is what we mean by zero to first order.)

Instead of considering a line, you can consider the level set $\{x : f(x) = f(x_0)\}$. So long as $x_0$ isn't a critical point of $f$, this level set will (at least in some neighbourhood of $x_0$) be a curve. Thus $f$ is constant along a curve in the direction $v$.

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Let's break our understanding in several pieces and glue them together to understand the puzzle:

  1. $f(x,y)$ is a number.
  2. $\nabla f(x,y)$ is a vector.
  3. $\vec{v}$ is also a vector.
  4. The directional derivative is a number that measures increase or decrease if you consider points in the direction given by $\vec{v}$.
  5. Therefore if $\nabla f(x,y) \cdot \vec{v} = 0$ then nothing happens. The function does not increase (nor decrease) when you consider points in the direction of $\vec{v}$.
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Let's start with the simplest case first. $f: \mathbb{R} \to \mathbb{R}$ given by $f(x) = x^2$. Then the directional derivative

$$\frac{\partial f}{\partial x} = \frac{df}{dx} = 2x$$

which is zero at the origin. If you look at a graph of a parabola, you see that the closer you zoom in on the origin, the more flat the graph looks (check this yourself on an online graph calculator).

Now take $g: \mathbb{R}^2 \to \mathbb{R}$ to be $g(x,y) = x^2 + y$, which is a parabolic cylinder, as can be seen here: http://www.wolframalpha.com/share/clip?f=d41d8cd98f00b204e9800998ecf8427empntaacb4q.

If you take the directional derivative along $y$ (also known as the partial derivative with respect to $y$), you get $1$. Thus, this function is increasing as you fix $x$ and increase $y$. However, the directional derivative along $x$ is $2x$, which is zero at the origin. Thus, if you zoom in really close to the origin, what you will see is something that looks like a tilted plane. As you run along the $x$ axis, the plane stays at the same height (constant), but it changes as you run along the $y$ axis.

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Suppose that $f$ is differentiable and that the directional derivative of $f$ along the vector $v=(a_1,a_2,...,a_n)$ is zero i.e. $\nabla f\cdot v=0$ or $a_1f_1+a_2f_2+...+a_nf_n=0$.

Claim: $f$ is constant along any line having direction $v$

$L$ be any such line which passes through $p=(b_1,b_2,...,b_n)$. Now the parametric equations of $L$ are: $x_1(s)=b_1+a_1s,x_2(s)=b_2+a_2s,...,x_n(s)=b_n+a_ns$. Keeping this in mind, we define $F(s)=f(x_1(s),x_2(s),...,x_n(s))$. The derivative $F'(s)=f_1\dot x_1(s)+f_2\dot x_2(s)+...+f_n\dot x_n(s)=f_1a_1+f_2a_2+...+f_na_n=0$. So $F$ must be a constant function. This shows that $f$ is constant along $L$.

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  • $\begingroup$ Why does $F'(s) = 0$ implies that $F$ is constant? $\endgroup$ – John Mars Apr 7 at 20:49
  • $\begingroup$ @JohnMars By Lagrange's mean value theorem, $F(x)-F(y)=F'(z)(x-y)=0$ for all $x,y$. So $F$ is constant. $\endgroup$ – Hrit Roy Apr 8 at 23:13

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