- #1
Rasalhague
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Lee: Introduction to Smooth Manifolds, definition A.18:
He then shows, by the chain rule, that
[tex]D_vf(a_0)= \sum_{i=1}^n v^i \frac{\partial }{\partial x^i}f(a) \bigg|_{a_0}[/tex]
It seems to me, though, that this number depends not only on the direction of [itex]v[/itex] but also on its length. For example if [itex]f(x,y,z) = xyz[/itex], and [itex]v=(1,0,0)[/itex], then
[tex]D_vf(2,3,4) = \begin{pmatrix}yz & xz & xy
\end{pmatrix} \bigg|_{(2,3,4)+0(1,0,0)} \begin{pmatrix}1\\0\\0\end{pmatrix} = 12.[/tex]
But if [itex]w=(2,0,0)[/itex], then the directional derivative of [itex]f[/itex] "in the direction of [itex]w[/itex]" (which is the same direction as the direction of [itex]v[/itex]) will be
[tex]D_vf(2,3,4) = \begin{pmatrix}yz & xz & xy
\end{pmatrix} \bigg|_{(2,3,4)+0(1,0,0)} \begin{pmatrix}2\\0\\0\end{pmatrix} = 24.[/tex]
So how does the definition make sense for any vector? What am I missing here?
Now suppose [itex]f : U \rightarrow \mathbb{R}[/itex] is a smooth real-valued function on an open set [itex]U \subseteq R^n[/itex], and [itex]a \in U[/itex]. For any vector [itex]v \in \mathbb{R}^n[/itex], we define the directional derivative of [itex]f[/itex] in the direction [itex]v[/itex] at [itex]a[/itex] to be the number
[tex]D_vf(a)=\frac{\mathrm{d} }{\mathrm{d} t} \bigg|_0 f(a+vt). \enspace\enspace(A.18)[/tex]
(This denition makes sense for any vector v; we do not require v to be a unit vector as one sometimes does in elementary calculus.)
He then shows, by the chain rule, that
[tex]D_vf(a_0)= \sum_{i=1}^n v^i \frac{\partial }{\partial x^i}f(a) \bigg|_{a_0}[/tex]
It seems to me, though, that this number depends not only on the direction of [itex]v[/itex] but also on its length. For example if [itex]f(x,y,z) = xyz[/itex], and [itex]v=(1,0,0)[/itex], then
[tex]D_vf(2,3,4) = \begin{pmatrix}yz & xz & xy
\end{pmatrix} \bigg|_{(2,3,4)+0(1,0,0)} \begin{pmatrix}1\\0\\0\end{pmatrix} = 12.[/tex]
But if [itex]w=(2,0,0)[/itex], then the directional derivative of [itex]f[/itex] "in the direction of [itex]w[/itex]" (which is the same direction as the direction of [itex]v[/itex]) will be
[tex]D_vf(2,3,4) = \begin{pmatrix}yz & xz & xy
\end{pmatrix} \bigg|_{(2,3,4)+0(1,0,0)} \begin{pmatrix}2\\0\\0\end{pmatrix} = 24.[/tex]
So how does the definition make sense for any vector? What am I missing here?