$ \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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A multivariable function is differentiable if there exist some linear map $T(\vec{h})$ such that $$\begin{align*} \lim_{\vec{h}\to\vec{0}}\frac{f(\vec{a}+\vec{h})-f(\vec{a})-T(\vec{h})}=0 \end{align*}$$

Concepts

In the one-dimensional case, the derivative tends to be defined as $$\begin{align*} f'(a)=\lim_{h\to 0} \frac{f(a+h)-f(h)}{|h|}. \end{align*}$$ Notice how we can rearrage this into $$\begin{align*} \frac{hf'(a)}{h}&=\lim_{h\to 0}\frac{f(a+h)-f(h)}{|h|}\\ \implies 0 &=\lim_{h\to 0} \frac{f(a+h)-f(h)-hf'(a)}{|h|} \end{align*}$$ The linear function $T(\vec{h})$ is basically the multivariable version of $hf'(a)$.

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A multivariable function is differentiable if there exist some linear map $T(\vec{h})$ such that $$\begin{align*} \lim_{\vec{h}\to\vec{0}}\frac{f(\vec{a}+\vec{h})-f(\vec{a})-T(\vec{h})}=0 \end{align*}$$

Concepts

In the one-dimensional case, the derivative tends to be defined as $$\begin{align*} f'(a)=\lim_{h\to 0} \frac{f(a+h)-f(h)}{|h|}. \end{align*}$$ Notice how we can rearrage this into $$\begin{align*} \frac{hf'(a)}{h}&=\lim_{h\to 0}\frac{f(a+h)-f(h)}{|h|}\\ \implies 0 &=\lim_{h\to 0} \frac{f(a+h)-f(h)-hf'(a)}{|h|} \end{align*}$$ The linear function $T(\vec{h})$ is basically the multivariable version of $hf'(a)$.

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definition
concepts
used in
hypothesis
results
FullPage
definition
concepts
used in
hypothesis
results
A multivariable function is differentiable if there exist some linear map $T(\vec{h})$ such that $$\begin{align*} \lim_{\vec{h}\to\vec{0}}\frac{f(\vec{a}+\vec{h})-f(\vec{a})-T(\vec{h})}=0 \end{align*}$$

Concepts

In the one-dimensional case, the derivative tends to be defined as $$\begin{align*} f'(a)=\lim_{h\to 0} \frac{f(a+h)-f(h)}{|h|}. \end{align*}$$ Notice how we can rearrage this into $$\begin{align*} \frac{hf'(a)}{h}&=\lim_{h\to 0}\frac{f(a+h)-f(h)}{|h|}\\ \implies 0 &=\lim_{h\to 0} \frac{f(a+h)-f(h)-hf'(a)}{|h|} \end{align*}$$ The linear function $T(\vec{h})$ is basically the multivariable version of $hf'(a)$.

Used In

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Hypothesis

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Results

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A multivariable function is differentiable if there exist some linear map $T(\vec{h})$ such that $$\begin{align*} \lim_{\vec{h}\to\vec{0}}\frac{f(\vec{a}+\vec{h})-f(\vec{a})-T(\vec{h})}=0 \end{align*}$$

Concepts

In the one-dimensional case, the derivative tends to be defined as $$\begin{align*} f'(a)=\lim_{h\to 0} \frac{f(a+h)-f(h)}{|h|}. \end{align*}$$ Notice how we can rearrage this into $$\begin{align*} \frac{hf'(a)}{h}&=\lim_{h\to 0}\frac{f(a+h)-f(h)}{|h|}\\ \implies 0 &=\lim_{h\to 0} \frac{f(a+h)-f(h)-hf'(a)}{|h|} \end{align*}$$ The linear function $T(\vec{h})$ is basically the multivariable version of $hf'(a)$.

Used In

Coming soon

Hypothesis

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Results

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FullPage
definition
concepts
used in
hypothesis
results