Examples:: *[[Examples]]*
Nonexamples:: *[[Nonexamples]]*
Constructions:: *[[Constructions|Used in the construction of...]]*
Specializations:: *[[Specializations]]*
Generalizations:: [[Euclidean diffeomorphism]]
Justifications and Intuition:: *[[Justifications and Intuition]]*
Properties:: [[derivatives are unique]], [[existence of derivative guarantees existence of all directional derivatives]], [[derivative of vector-valued function is vector of component derivatives]], [[total derivative equals Jacobian]]
Sufficiencies:: *[[Sufficiencies]]*
Equivalences:: *[[Equivalences]]*
> [!definition]
> Let $X,Y$ be [[Banach space|Banach spaces]] over $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$, $\Omega \subset X$ open. Let $p \in \Omega$. A function $F:\Omega \to Y$ is in general not [[linear map|linear]]; one cannot expect identities like $F(x+h)\neq F(x)+F(h)$. We say $F$ is **(Fréchet) differentiable at $p \in \Omega$** if near $p$ there is a linear approximation: [[operator norm|there exists]] $DF |_{p} \in \text{Hom}_{\mathsf{Ban_{T}}}(X,Y)$ [[asymptotic notation|satisfying]] $F(p+h)=F(p)+ DF |_{p}(h)+o(\|h\|)$
> in a [[neighborhood]] of $p$. [[asymptotic notation|In other words]], $\|F(p+h)-F(p)- DF |_{p}(h)\|=o(\|h\|)$: the error grows strictly slower than $\|h\|$. Equivalently, we can write $\lim_{\|h\| \to 0}\frac{\|F(p+h)-F(p)-DF |_{p}(h)\|}{\|h\|}=0.$
> $DF |_{p}(h)$ is unique if it exists.
>
> We say $F$ is **(Fréchet) differentiable** if it is so at every $p \in \Omega$, and call the (possibly [[continuous|discontinuous]]) map $DF:\Omega \to \text{Hom}_{\mathsf{Ban_{T}}}(X,Y)$
> the **(Fréchet) derivative** of $F$. (If $DF$ is [[continuous]] we say $F$ is [[continuously differentiable]] and write $F \in C^{1}(\Omega)$.)
>
> When $\mathbb{F}=\mathbb{C}$, it is common to use the term **holomorphic** instead of (Fréchet) differentiable. Holomorphic functions have a huge amount of extra structure and [[holomorphic|deserve their own note]].
----
> [!specialization]
>
> - Let $\Omega$ be an [[open set]] in $\mathbb{R}^{m}$; let $f:\mathbb{R} ^{m} \to \mathbb{R}^{n}$.
>
> > [!definition] Definition. (Derivative of $f:\Omega \subset \mathbb{R}^{m} \to \mathbb{R}^{n}$)
> > Suppose $\vec f: \Omega \subset \mathbb{R}^m \to \mathbb{R}^n$ with $\Omega$ an [[open set]] in $\mathbb{R}^m$. Let $\vec p \in \Omega$. We say **"$\vec f$ is differentiable at $\vec p
quot;** if there exists $B \in \mathbb{R}^{n \times m}$ such that $\lim_{ \vec h \to \vec 0 } \frac{ \big\vert \vec f(\vec p+ \vec h) - \vec f(\vec p) - B\vec h \big\vert}{\big\vert \vec h \big\vert}=0.$
> If such a $B$ exists, it is unique, and $B$ is called the [[derivative]] of $\vec f$ at $\vec p$, written $B=D \vec f (\vec p)$.
> >
>
>
>
>
>
> > [!intuition] Motivation.
> > # Single-variable definition from calculus
> Let $\phi : \Omega \subset \mathbb{R} \to \mathbb{R}$ with $\Omega$ open in $\mathbb{R}$, let $p \in \Omega$.
> >
> To say "$\phi$ is **differentiable** at $pquot; is to assert that the following [[function limit]] exists: $\lim_{ t \to 0 } \frac{\phi(p+t) - \phi(p)}{(p+t) - p}, $ and we call that limit the [[derivative]] of $\phi$ at $p$. Unfortunately, this definition *breaks down* if we attempt to naively generalize it to functions of several variables.
> >
> > # Towards a [[vector]]-valued definition
> (must have read above section) *so what do we do?* First, rewrite the single-variable definition from calculus: $\lim_{ t \to 0 } \frac{\phi(p+t) - \phi(p)}{t}$ which exists if and only if *there exists $b \in \mathbb{R}$ such that*$\lim_{ t \to 0 } \big\vert \frac{\phi(p+t) - \phi(p)}{t} - b \big\vert = 0,$
> which is an equivalent expression to saying that (make common denominator) *there exists $b \in \mathbb{R}$ such that* $\lim_{ t \to 0 } \big\vert \frac{\phi(p+t) - \phi(p) - bt}{t} \big\vert = 0,$
> i.e., *there exists $b \in \mathbb{R}$ such that* $\lim_{ t \to 0 } \frac{ \big\vert \phi(p+t) - \phi(p) - bt \big\vert}{\big\vert t \big\vert} = 0.$
> Now **this** expression looks like one we can better generalize, replacing [[absolute value]]s with [[metric]]s and the like...
> ># Derivative of a [[vector]]-valued function
> > **Note. Now, $\vert \cdot \vert$ denotes the [[sup norm]], NOT [[absolute value]]!**
> > #### Remark.
> > We can actually remove the [[sup norm]] in the numerator above and keep an equivalent statement, that is, the above holds **iff** $\lim_{ \vec h \to \vec 0 } \frac{ \vec f(\vec p+ \vec h) - \vec f(\vec p) - B\vec h }{\big\vert \vec h \big\vert}=\vec 0.$
>
>
> ### Intuition
> Don't forget that, in the single-variable case, the $b$ from whence $B$ came in the above 'derivation' was originally the value which, when subtracted from $\lim_{ t \to 0 } \frac{\phi(p+t) - \phi(p)}{t}$, yielded 0, i.e., $\lim_{ t \to 0 } \vert \frac{\textcolor{Skyblue}{\phi(p+t) - \phi(p)} - \textcolor{Thistle}{bt}}{t} \vert=0$. We called $b$ the derivative of $\phi$ at $p$, since it is the value that makes equality to 0 hold. But here we're looking at the notion of a *best linear approximation* $\textcolor{Thistle}{\text{err(t)} = bt}$ to the 'incrementer function' $\textcolor{Skyblue}{\phi(p+t) - \phi(p)}$, also called the *best first-order approximation*. But a *best linear function* is an idea that comes straight from linear algebra — [[linear map]]s and [[matrix|matrices]]! That's why we generalize $b$ to $B$.
>
> Another way to look at things would be that it happens to be the case that $b$ is also the [[function limit]] of the difference quotient. Therefore, it should make sense that we require $B$ to yield equality to 0 in a similar way; it may not be the limit of anything *per se* (at least i don't think?), but it behaves the same way as the single-variable derivative in a way that we've decided is more useful to care about!
>
>
# How do we find $B$?
- [[total derivative equals Jacobian]]
> [!backlink]
> ```dataview
> TABLE rows.file.link as "Further Reading"
> FROM [[]]
> FLATTEN file.tags as Tag
> WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom"
> GROUP BY Tag
> ```
> [!frontlink]
> ```dataview
> TABLE rows.file.link as "Further Reading"
> FROM outgoing([[]])
> FLATTEN file.tags as Tag
> WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom"
> GROUP BY Tag
> ```