Title: Theory / Important / Math / Analysis / Complex / Visual+21 / 04 Differentiation/
Tags: examplesSwitchExample1 counterExample

Name: The image of \(\overrightarrow{qp}\) by \(f\)
Definition: \(\overrightarrow{f(q)f(p)}\)
Page: 192
Chapter: 4I
Section: 3
Variables: \(q,p\in\mathbb C\),
\(f:\mathbb{C\to C}\)

Definition: [$]\pdv{ }{x}[/$]
Page: 192
Chapter: 4III
Section: 2
Notation: \(\partial_x\)

Definition: [$]\pdv{u}{x}\dd x+\pdv{u}{y}\dd y[/$]
Page: 192
Chapter: 4III
Section: 2
Notation: [$]\dd u[/$]

Name: The Jacobian matrix of \(z\)
Definition: \(\left(\begin{array}{ll}\partial xu&\partial yu\\\partial xv&\partial yv\end{array}\right)\)
Page: 193
Chapter: 4III
Section: 2
Variables: \(u,v:\mathbb C\to\mathbb R\),
\(z=u+iv\)
Example: \(2r\left(\begin{array}{ll}\cos\theta&-\sin\theta\\\sin \theta&\cos\theta\end{array}\right)\)
Denoted by: \(J\)
Applied to: \(z^2\) at \(R\angle\theta\)

Name: A conformal function
Page: 190
Chapter: 4II
Intuition: A function preserving angles

Name: A function which amplifies the vectors at \(q\)
Definition: A function such that [$]\abs{\dd\overrightarrow{QP}}=c\abs{\dd\overrightarrow{qp}}[/$], for each \(p\) near \(q\) and \(c\) a real constant
Page: 193
Chapter: 4III
Section: 3
Variables: \(q\) a complex

Name: A function which twists the vectors at \(q\)
Definition: A function such that [$]\arg\dd \overrightarrow{QP}=\theta\cdot\arg\dd \overrightarrow{qp}[/$], for each \(p\) near \(q\) and \(\theta\) a real
Page: 193
Chapter: 4III
Section: 3
Variables: \(q\) a complex

Name: The amplication of \(f\) at \(q\)
Definition: the constant \(c\) such that [$]\abs{\dd\overrightarrow{QP}}=c\abs{\dd \overrightarrow{qp}}[/$], for each \(p\) near \(q\)
Page: 193
Chapter: 4III
Section: 3
Variables: \(q\) complex
\(f\) function which amplifies the vectors at \(q\)
Example: is \(2r\)
Applied to: The amplification of \(z\mapsto z^2\) at \(re^{i\theta}\)

Name: The twist of \(f\) at \(q\)
Definition: the constant \(\theta\) such that [$]\arg\dd\overrightarrow{QP}=\theta\cdot\arg\dd\overrightarrow{qp}[/$], for each \(p\) near \(q\)
Page: 193, 194
Chapter: 4III
Section: 3
Variables: \(q\) complex
\(f\) function which twists the vectors at \(q\)
Example: \(\theta\)
Applied to: \(z\mapsto z^2\) at \(re^{i\theta}\)

Name: An amplitwist at \(q\)
Definition: An amplification and a twist at \(q\)
Page: 194
Chapter: 4III
Section: 3
Variables: \(q\) a complex
Example: \(z\mapsto z^2\) is an amplitwist at \(re^{i\theta}\) with amplification \(2r\) and twist \(\theta\)

Smaller0: \(f\) is an amplitwist
Greater0: \(f\) is conformal
Page: 194
Chapter: 4III
Section: 3
Variables: \(f\) a complex function