$ \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}} $
FullPage
result
concepts
hypothesis
implications
proof

Relation between derivatives of differentable point

Let $S\subseteq\mathbb{R}^n$, $\vec{a}\in \text{int}(A)$, and $f:A\to\mathbb{R}^m$. Suppose $f$ is differentiable at $\vec{a}$ and let $T=Df(\vec{a})$ be the derivative of $f$ at $\vec{a}$. Then:
  1. For every unit vector $\vec{u}\in \mathbb{R}^n$,the directional derivative of $f$ at $\vec{a}$ in the direction $\vec{u}$ exists and is $D_{\vec{u}}f(\vec{a})=T(\vec{u})$.
  2. All partial derivatives $\frac{\partial f_i}{\partial x_j}(\vec{a})$, $1\leq i\leq m$, $1\leq j\leq n$, exists.
  3. The $m\times n$ matrix representing $T$ in the standard basis is the Jacobian matrix $$\begin{align*} J=\begin{bmatrix}\partial{\partial f_1}{\partial x_1}(\vec{a}) & \dots & \frac{\partial f_1}{\partial x_n}(\vec{a}) \\ \vdots & \ddots & \vdots \\ \frac{\partial f_m}{\partial x_1}(\vec{a})& \dots & \frac{\partial f_m}{\partial x_n}(\vec{a}) \end{bmatrix} \end{align*}$$

Concepts

Coming soon

Hypothesis

Coming soon

Results

Coming soon

Proof

Coming soon

Relation between derivatives of differentable point

Let $S\subseteq\mathbb{R}^n$, $\vec{a}\in \text{int}(A)$, and $f:A\to\mathbb{R}^m$. Suppose $f$ is differentiable at $\vec{a}$ and let $T=Df(\vec{a})$ be the derivative of $f$ at $\vec{a}$. Then:
  1. For every unit vector $\vec{u}\in \mathbb{R}^n$,the directional derivative of $f$ at $\vec{a}$ in the direction $\vec{u}$ exists and is $D_{\vec{u}}f(\vec{a})=T(\vec{u})$.
  2. All partial derivatives $\frac{\partial f_i}{\partial x_j}(\vec{a})$, $1\leq i\leq m$, $1\leq j\leq n$, exists.
  3. The $m\times n$ matrix representing $T$ in the standard basis is the Jacobian matrix $$\begin{align*} J=\begin{bmatrix}\partial{\partial f_1}{\partial x_1}(\vec{a}) & \dots & \frac{\partial f_1}{\partial x_n}(\vec{a}) \\ \vdots & \ddots & \vdots \\ \frac{\partial f_m}{\partial x_1}(\vec{a})& \dots & \frac{\partial f_m}{\partial x_n}(\vec{a}) \end{bmatrix} \end{align*}$$

Concepts

Coming soon

Hypothesis

Coming soon

Results

Coming soon

Proof

Coming soon
FullPage
result
concepts
hypothesis
implications
proof
FullPage
result
concepts
hypothesis
implications
proof

Relation between derivatives of differentable point

Let $S\subseteq\mathbb{R}^n$, $\vec{a}\in \text{int}(A)$, and $f:A\to\mathbb{R}^m$. Suppose $f$ is differentiable at $\vec{a}$ and let $T=Df(\vec{a})$ be the derivative of $f$ at $\vec{a}$. Then:
  1. For every unit vector $\vec{u}\in \mathbb{R}^n$,the directional derivative of $f$ at $\vec{a}$ in the direction $\vec{u}$ exists and is $D_{\vec{u}}f(\vec{a})=T(\vec{u})$.
  2. All partial derivatives $\frac{\partial f_i}{\partial x_j}(\vec{a})$, $1\leq i\leq m$, $1\leq j\leq n$, exists.
  3. The $m\times n$ matrix representing $T$ in the standard basis is the Jacobian matrix $$\begin{align*} J=\begin{bmatrix}\partial{\partial f_1}{\partial x_1}(\vec{a}) & \dots & \frac{\partial f_1}{\partial x_n}(\vec{a}) \\ \vdots & \ddots & \vdots \\ \frac{\partial f_m}{\partial x_1}(\vec{a})& \dots & \frac{\partial f_m}{\partial x_n}(\vec{a}) \end{bmatrix} \end{align*}$$

Concepts

Coming soon

Hypothesis

Coming soon

Results

Coming soon

Proof

Coming soon

Relation between derivatives of differentable point

Let $S\subseteq\mathbb{R}^n$, $\vec{a}\in \text{int}(A)$, and $f:A\to\mathbb{R}^m$. Suppose $f$ is differentiable at $\vec{a}$ and let $T=Df(\vec{a})$ be the derivative of $f$ at $\vec{a}$. Then:
  1. For every unit vector $\vec{u}\in \mathbb{R}^n$,the directional derivative of $f$ at $\vec{a}$ in the direction $\vec{u}$ exists and is $D_{\vec{u}}f(\vec{a})=T(\vec{u})$.
  2. All partial derivatives $\frac{\partial f_i}{\partial x_j}(\vec{a})$, $1\leq i\leq m$, $1\leq j\leq n$, exists.
  3. The $m\times n$ matrix representing $T$ in the standard basis is the Jacobian matrix $$\begin{align*} J=\begin{bmatrix}\partial{\partial f_1}{\partial x_1}(\vec{a}) & \dots & \frac{\partial f_1}{\partial x_n}(\vec{a}) \\ \vdots & \ddots & \vdots \\ \frac{\partial f_m}{\partial x_1}(\vec{a})& \dots & \frac{\partial f_m}{\partial x_n}(\vec{a}) \end{bmatrix} \end{align*}$$

Concepts

Coming soon

Hypothesis

Coming soon

Results

Coming soon

Proof

Coming soon
FullPage
result
concepts
hypothesis
implications
proof