Let-f-D-f-R-n-R-m-let-a-be-an-interior-point-of-Dom-f-and-let-u-be-any-vector-in-R-n-when-is-a-vector-v-R-m-called-the-directional-derivative-of-f-at-a-along-the-line-determine-by-u-he

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Question Number 192397 by Mastermind last updated on 16/May/23

$$\mathrm{Let}\mathrm{f}:\mathrm{D}\left(\mathrm{f}\right)\subseteq {\mathbb{R}}^{\mathrm{n}}\to {\mathbb{R}}^{\mathrm{m}}$$$$\mathrm{let}{}^{\prime }{\mathrm{a}}^{\prime }\mathrm{be}\mathrm{an}\mathrm{interior}\mathrm{point}\mathrm{of}\mathrm{Dom}\left(\mathrm{f}\right)$$$$\mathrm{and}\mathrm{let}{}^{\prime }{\mathrm{u}}^{\prime }\mathrm{be}\mathrm{any}\mathrm{vector}\mathrm{in}{\mathbb{R}}^{\mathrm{n}},\mathrm{when}$$$$\mathrm{is}\mathrm{a}\mathrm{vector}\mathrm{v}\in {\mathbb{R}}^{\mathrm{m}}\mathrm{called}\mathrm{the}\mathrm{directional}$$$$\mathrm{derivative}\mathrm{of}\mathrm{f}\mathrm{at}{}^{\prime }{\mathrm{a}}^{\prime }\mathrm{along}\mathrm{the}\mathrm{line}$$$$\mathrm{determine}\mathrm{by}\mathrm{u}?$$$$$$$$\mathrm{help}!$$

Answered by aleks041103 last updated on 21/May/23

$$let{f}_i:D\left(f\right)\subseteq {\mathbb{R}}^n\to \mathbb{R}bethecomponent$$$$functionsoff,i.e.$$$$f\left(a\right)=(f_1\left(a\right),f_2\left(a\right),\dots ,f_m\left(a\right))\in {\mathbb{R}}^m$$$$\Rightarrow inthiscase:$$$$v=(\frac{\partial f_1}{\partial u}\left(a\right),\frac{\partial f_2}{\partial u}\left(a\right),\dots ,\frac{\partial f_m}{\partial u}\left(a\right))\in {\mathbb{R}}^m$$$$where\frac{\partial }{\partial u}istheordinarydirectional$$$$derivtive,i.e.\frac{\partial }{\partial u}=u\cdot grad=u\cdot ▽$$$$\Rightarrow v=((u\cdot grad\left(f_1\right))\left(a\right),(u\cdot grad\left(f_2\right))\left(a\right),\dots ,(u\cdot grad\left(f_m\right))\left(a\right))\in {\mathbb{R}}^m$$$$inindexnotation:$$$$v_k=\underset{s=1}{\sum ^n}{u}_s\frac{\partial f_k}{\partial x_s}\left(a\right)$$$$orusingeinsteinnotation$$$$v_k=u_s{\partial }_s{f}_k\left(a\right)$$$$orinvectornotation$$$$v=\left((u\cdot ▽)f\right)\left(a\right)$$

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