$ \newcommand{\braket}[1]{\langle #1 \rangle} \newcommand{\abs}[2][]{\left\lvert#2\right\rvert_{\text{#1}}} \newcommand{\ket}[1]{\left\lvert#1 \right.\rangle} \newcommand{\bra}[1]{\langle\left. #1\right\rvert} \newcommand{\braket}[1]{\langle #1 \rangle} \newcommand{\dd}{\text{d}} \newcommand{\dv}[2]{\frac{\dd #1}{\dd #2}} \newcommand{\pdv}[2]{\frac{\partial}{\partial #1}} $
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Components of Directional derivatives

Suppose $A\subseteq\mathbb{R}^n$, $\vec{a}\in\text{int}(A)$, and $\vec{u}\in \mathbb{R}^n$ be a unit vector. Let $f:A\to\mathbb{R}^m$ be a function with components $f_i:A\to\mathbb{R}$, $i=1, 2, \dots, m$. Then $D_{\vec{u}}f(\vec{a})$ exists if and if $D_{\vec{u}}f_i(\vec{a})$ exists for each $i=1, 2, \dots, m$. Furthermore, $$\begin{align*} D_{\vec{u}}f(\vec{a})=(D_{\vec{u}}f_1(\vec{a}), D_{\vec{u}}f_2(\vec{a}), \dots, D_{\vec{u}}f(\vec{a})). \end{align*}$$

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Components of Directional derivatives

Suppose $A\subseteq\mathbb{R}^n$, $\vec{a}\in\text{int}(A)$, and $\vec{u}\in \mathbb{R}^n$ be a unit vector. Let $f:A\to\mathbb{R}^m$ be a function with components $f_i:A\to\mathbb{R}$, $i=1, 2, \dots, m$. Then $D_{\vec{u}}f(\vec{a})$ exists if and if $D_{\vec{u}}f_i(\vec{a})$ exists for each $i=1, 2, \dots, m$. Furthermore, $$\begin{align*} D_{\vec{u}}f(\vec{a})=(D_{\vec{u}}f_1(\vec{a}), D_{\vec{u}}f_2(\vec{a}), \dots, D_{\vec{u}}f(\vec{a})). \end{align*}$$

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Components of Directional derivatives

Suppose $A\subseteq\mathbb{R}^n$, $\vec{a}\in\text{int}(A)$, and $\vec{u}\in \mathbb{R}^n$ be a unit vector. Let $f:A\to\mathbb{R}^m$ be a function with components $f_i:A\to\mathbb{R}$, $i=1, 2, \dots, m$. Then $D_{\vec{u}}f(\vec{a})$ exists if and if $D_{\vec{u}}f_i(\vec{a})$ exists for each $i=1, 2, \dots, m$. Furthermore, $$\begin{align*} D_{\vec{u}}f(\vec{a})=(D_{\vec{u}}f_1(\vec{a}), D_{\vec{u}}f_2(\vec{a}), \dots, D_{\vec{u}}f(\vec{a})). \end{align*}$$

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Components of Directional derivatives

Suppose $A\subseteq\mathbb{R}^n$, $\vec{a}\in\text{int}(A)$, and $\vec{u}\in \mathbb{R}^n$ be a unit vector. Let $f:A\to\mathbb{R}^m$ be a function with components $f_i:A\to\mathbb{R}$, $i=1, 2, \dots, m$. Then $D_{\vec{u}}f(\vec{a})$ exists if and if $D_{\vec{u}}f_i(\vec{a})$ exists for each $i=1, 2, \dots, m$. Furthermore, $$\begin{align*} D_{\vec{u}}f(\vec{a})=(D_{\vec{u}}f_1(\vec{a}), D_{\vec{u}}f_2(\vec{a}), \dots, D_{\vec{u}}f(\vec{a})). \end{align*}$$

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