$ \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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the Jacobian matrix is the $m\times n$ matrix representing the multivariable derivative T=Df(\vec{a}) in the standard basis, and has elements $J_{i,j}=[\frac{\partial f_i}{\partial x_j}]$, giving $$\begin{align*} J:=\begin{bmatrix}\frac{\partial f_1}{\partial x_1}(\vec{a}) & \dots & \frac{\partial f_1}{\partial x_n}(\vec{a}) \\ \vdots & \ddots & \vdots \\ \frac{\partial f_m}{\partial x_1}(\vec{a}) & \dots & \frac{\partial f_m}{\partial x_n}(\vec{a})\end{bmatrix} \end{align*}$$

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There's a proof for why this is. It's also related to a theorem about differentiable functions.

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the Jacobian matrix is the $m\times n$ matrix representing the multivariable derivative T=Df(\vec{a}) in the standard basis, and has elements $J_{i,j}=[\frac{\partial f_i}{\partial x_j}]$, giving $$\begin{align*} J:=\begin{bmatrix}\frac{\partial f_1}{\partial x_1}(\vec{a}) & \dots & \frac{\partial f_1}{\partial x_n}(\vec{a}) \\ \vdots & \ddots & \vdots \\ \frac{\partial f_m}{\partial x_1}(\vec{a}) & \dots & \frac{\partial f_m}{\partial x_n}(\vec{a})\end{bmatrix} \end{align*}$$

Concepts

There's a proof for why this is. It's also related to a theorem about differentiable functions.

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definition
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FullPage
definition
concepts
used in
hypothesis
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the Jacobian matrix is the $m\times n$ matrix representing the multivariable derivative T=Df(\vec{a}) in the standard basis, and has elements $J_{i,j}=[\frac{\partial f_i}{\partial x_j}]$, giving $$\begin{align*} J:=\begin{bmatrix}\frac{\partial f_1}{\partial x_1}(\vec{a}) & \dots & \frac{\partial f_1}{\partial x_n}(\vec{a}) \\ \vdots & \ddots & \vdots \\ \frac{\partial f_m}{\partial x_1}(\vec{a}) & \dots & \frac{\partial f_m}{\partial x_n}(\vec{a})\end{bmatrix} \end{align*}$$

Concepts

There's a proof for why this is. It's also related to a theorem about differentiable functions.

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Hypothesis

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the Jacobian matrix is the $m\times n$ matrix representing the multivariable derivative T=Df(\vec{a}) in the standard basis, and has elements $J_{i,j}=[\frac{\partial f_i}{\partial x_j}]$, giving $$\begin{align*} J:=\begin{bmatrix}\frac{\partial f_1}{\partial x_1}(\vec{a}) & \dots & \frac{\partial f_1}{\partial x_n}(\vec{a}) \\ \vdots & \ddots & \vdots \\ \frac{\partial f_m}{\partial x_1}(\vec{a}) & \dots & \frac{\partial f_m}{\partial x_n}(\vec{a})\end{bmatrix} \end{align*}$$

Concepts

There's a proof for why this is. It's also related to a theorem about differentiable functions.

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Hypothesis

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