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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
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 ? help!
$$\mathrm{Let}\:\mathrm{f}:\mathrm{D}\left(\mathrm{f}\right)\subseteq\mathbb{R}^{\mathrm{n}} \rightarrow\mathbb{R}^{\mathrm{m}} \\ $$$$\mathrm{let}\:'\mathrm{a}'\:\mathrm{be}\:\mathrm{an}\:\mathrm{interior}\:\mathrm{point}\:\mathrm{of}\:\mathrm{Dom}\left(\mathrm{f}\right) \\ $$$$\mathrm{and}\:\mathrm{let}\:'\mathrm{u}'\:\mathrm{be}\:\mathrm{any}\:\mathrm{vector}\:\mathrm{in}\:\mathbb{R}^{\mathrm{n}} ,\:\mathrm{when} \\ $$$$\mathrm{is}\:\mathrm{a}\:\mathrm{vector}\:\mathrm{v}\in\mathbb{R}^{\mathrm{m}} \:\mathrm{called}\:\mathrm{the}\:\mathrm{directional} \\ $$$$\mathrm{derivative}\:\mathrm{of}\:\mathrm{f}\:\mathrm{at}\:'\mathrm{a}'\:\mathrm{along}\:\mathrm{the}\:\mathrm{line} \\ $$$$\mathrm{determine}\:\mathrm{by}\:\mathrm{u}\:? \\ $$$$ \\ $$$$\mathrm{help}! \\ $$
Answered by aleks041103 last updated on 21/May/23
let f_i :D(f)⊆R^n →R be the component functions of f, i.e. f(a)=(f_1 (a),f_2 (a),...,f_m (a))∈R^m ⇒in this case: v=((∂f_1 /∂u)(a),(∂f_2 /∂u)(a),...,(∂f_m /∂u)(a))∈R^m where (∂/∂u) is the ordinary directional derivtive, i.e. (∂/∂u)=u∙grad=u∙▽ ⇒v=((u∙grad(f_1 ))(a),(u∙grad(f_2 ))(a),...,(u∙grad(f_m ))(a))∈R^m in index notation: v_k =Σ_(s=1) ^n u_s (∂f_k /∂x_s )(a) or using einstein notation v_k =u_s ∂_s f_k (a) or in vector notation v=((u∙▽)f)(a)
$${let}\:{f}_{{i}} :{D}\left({f}\right)\subseteq\mathbb{R}^{{n}} \rightarrow\mathbb{R}\:{be}\:{the}\:{component} \\ $$$${functions}\:{of}\:{f},\:{i}.{e}. \\ $$$${f}\left({a}\right)=\left({f}_{\mathrm{1}} \left({a}\right),{f}_{\mathrm{2}} \left({a}\right),…,{f}_{{m}} \left({a}\right)\right)\in\mathbb{R}^{{m}} \\ $$$$\Rightarrow{in}\:{this}\:{case}: \\ $$$${v}=\left(\frac{\partial{f}_{\mathrm{1}} }{\partial{u}}\left({a}\right),\frac{\partial{f}_{\mathrm{2}} }{\partial{u}}\left({a}\right),…,\frac{\partial{f}_{{m}} }{\partial{u}}\left({a}\right)\right)\in\mathbb{R}^{{m}} \\ $$$${where}\:\frac{\partial}{\partial{u}}\:{is}\:{the}\:{ordinary}\:{directional} \\ $$$${derivtive},\:{i}.{e}.\:\frac{\partial}{\partial{u}}={u}\centerdot{grad}={u}\centerdot\bigtriangledown \\ $$$$\Rightarrow{v}=\left(\left({u}\centerdot{grad}\left({f}_{\mathrm{1}} \right)\right)\left({a}\right),\left({u}\centerdot{grad}\left({f}_{\mathrm{2}} \right)\right)\left({a}\right),…,\left({u}\centerdot{grad}\left({f}_{{m}} \right)\right)\left({a}\right)\right)\in\mathbb{R}^{{m}} \\ $$$${in}\:{index}\:{notation}: \\ $$$${v}_{{k}} =\underset{{s}=\mathrm{1}} {\overset{{n}} {\sum}}{u}_{{s}} \frac{\partial{f}_{{k}} }{\partial{x}_{{s}} }\left({a}\right) \\ $$$${or}\:{using}\:{einstein}\:{notation} \\ $$$${v}_{{k}} ={u}_{{s}} \partial_{{s}} {f}_{{k}} \left({a}\right) \\ $$$${or}\:{in}\:{vector}\:{notation} \\ $$$${v}=\left(\left({u}\centerdot\bigtriangledown\right){f}\right)\left({a}\right) \\ $$

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