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I am jammed to problem 11 on page 803 here. Earlier on page 797, it claims that the function changes the least when $\langle \nabla f, \bar{e}\rangle=0$ and it changes the most in the direction of the gradient $\nabla f$.

Exapmle

Suppose $f(x,y,z)=\bar{x}\bar{y}^{2}+\bar{y}\bar{z}^{3}$.

Slowest growth

$\begin{align*} \langle \nabla f, \bar{e}\rangle &= (\bar{x}\bar{y}^{2}+\bar{y}\bar{z}^{3})\cdot \bar{e}_{x}+ (\bar{x}\bar{y}^{2}+\bar{y}\bar{z}^{3})\cdot \bar{e}_{y}+ (\bar{x}\bar{y}^{2}+\bar{y}\bar{z}^{3})\cdot \bar{e}_{z}\\ &= \bar{y}^{2} x +(\bar{x} y^{2}+\bar{z}^{3} y)+(\bar{y}z^{3}) \\&= 0 \end{align*}$

so it is the direction of the slowest growth (according to my book -rule $\langle \nabla f, \bar{e}\rangle =0$, p797)?

Greatest growth direction

$\bar{n}=\max \frac{\nabla f}{\left| f \right|}$ so just I need to find out the inflection points $\nabla^{2} f$ so

$\nabla^{2} f=\begin{pmatrix}\bar{y}^{2}\\ 2\bar{x}\bar{y} \\ 3\bar{y} \bar{z}^{2} \end{pmatrix}=\bar{0},$

I feel I am terribly misunderstanding something.

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1 Answer 1

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$\DeclareMathOperator{\bR}{\mathbf R}$There does appear to be some confusion. Let $p$ be a point in $\bR^3$ and $f\colon \bR^3 \to \mathbf R^3$ a function. If $u$ is a unit vector in $\bR^3$, then we have the notion of the directional derivative of $f$ at $p$ in the direction of $u$, denoted $D_uf(p)$. Finding the direction of "fastest growth" is finding a $u$ which maximizes this quantity.

If $f$ is differentiable at $p$ then it isn't hard to show that $D_uf(p) = f(p) \cdot u$, where $\nabla f(p)$ is the gradient. Let's make sure that we can compute this: in your example, \[ \nabla f(x, y, z) = (y^2, 2xy + z^3, 3yz^3). \] Using the Schwarz inequality you can conclude that $D_uf(p)$ attains its maximum when $u$ points in the direction of $\nabla f(p)$. The minimum occurs when $u$ points in the opposite direction. If we specialize to $p = (1, 1, 1)$ in the example then $\nabla f(p) = (1, 3, 3)$, and $f$ grows quickest in the direction \[ u = \frac{1}{\sqrt{19}}(1, 3, 3). \]

However, "least change" means minimizing $|D_uf(a)|$. This magnitude is actually zero when $\nabla f(p) \cdot u = 0$, i.e. when $u$ lies in the plane (I should assume that $\nabla f(p) \neq 0$) through the origin with normal vector $\nabla f(p)$. So any unit vector in that plane (and there are infinitely many) will do. Continuing the example, I need to find a $u = (u_1, u_2, u_3)$ such that \[ \begin{pmatrix}1 & 3 & 3\end{pmatrix}\begin{pmatrix}u_1 \\ u_2 \\ u_3\end{pmatrix} = u_1 + 3u_2 + 3u_3 = 0. \] You could find a basis for this null space, or just notice that $(3, -1, 0)$ works and normalize this to get \[ u = \frac{1}{\sqrt{10}}(3, -1, 0). \]

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