Title: Theory / Important / Math / Analysis / Complex / Visual+21 / 10 Vector fields
Tags: examplesSwitchExample1

Name: The representation of \(f\) as a vector field.
Definition: The representation of \(f(z)\) as a vector, with starting point \(z\), for some \(z\) belonging to some subset of \(\mathbb C\)
for \(\frac1z\)
for \(z^2\)
Page: 450
Chapter: 10I
Section: 1
Variables: \(f:\mathbb{C\to C}\)

Name: The representation of \(z\) as a vector
Definition: An arrow showing the direction of \(z\), with length proportional to \(z\)
Page: 450
Chapter: 10I
Section: 1
Variables: \(z\in\mathbb C\)
Example: for \(2i+3\)

Name: A steady vector space
Definition: A vector space not varying with time
Page: 452
Chapter: 10I
Section: 1

Name: A line of force of \(f\)
Definition: The integral curve of \(f\)
Page: 453
Chapter: 10I
Section: 1
Variables: \(f:\mathbb{C\to C}\)
Assuming: \(f\) is a force field

Name: The phase portrait of \(f\)
Definition: Representing \(f\) using its streamlines
Page: 453
Chapter: 10I
Section: 1
Variables: \(f:\mathbb{C\to C}\)

Name: A line of force of \(f\)
Definition: An integral curve of \(f\)
Page: 453
Chapter: 10I
Section: 1
Variables: \(f:\mathbb{C\to C}\)
Assuming: \(f\) is a force field

Name: An integral curve of \(f\)
Definition: Some path where, at each point \(z\), the tangeant is the argument of \(f(z)\)
One of the line of
Page: 453
Chapter: 10I
Section: 1
Variables: \(f:\mathbb{C\to C}\)

Name: A flow
Definition: A vector field representing moving things (particles, liquid...)
Page: 454
Chapter: 10I
Section: 1

Name: A force field
Definition: A vector field representing force acting on points of the space
Page: 454
Chapter: 10I
Section: 1

Name: A source of strengh \(S\)
Definition: A point at which the function \(f\) pump in \(S\) units of fluid per unit of time.
Page: 454
Chapter: 10I
Section: 4
Variables: \(f:\mathbb{C\to C}\) a vector field

Name: A sink of strengh \(S\)
Definition: A point at which we pump out of \(S\) units of fluid per unit of time.
Page: 454
Chapter: 10I
Section: 4
Variables: \(f:\mathbb{C\to C}\) a vector field

Name: The flux of \(f\) through \(C\)
Definition: The integral of the elementary fluxes through \(C\)
Page: 454
Chapter: 10I
Section: 4
Variables: \(f:\mathbb{C\to C}\) a vector field
\(C\) a curve
Intuition: The amount of fluid flowing across \(C\) in each unit of time

Name: The elementary flux of \(f\) through \(e\)
Definition: The length [$]\dd l[/$] of the part of the curve, multiplied by the component of the velocity perpendicular to the curve.
Page: 454
Chapter: 10I
Section: 4
Variables: \(f:\mathbb{C\to C}\) a vector field
\(C\) a curve
\(e\in C\)

Name: A doublet
Definition: A source and a sink of equal strength
Page: 456
Chapter: 10I
Section: 4

Name: A singular point
Definition: A point \(p\) such that \(f(p)\) is either 0 or infinite
Page: 456
Chapter: 10II
Section: 1
Variables: \(f\) a vector field
Name2: A critical point (not used in this book)
Intuition: Intersection of different stream line (in phase portrait)
Name3: a singularity (not used in this book)

Name: A simple crosspoint
Page: 456
Chapter: 10II
Section: 1
Variables: \(V\) a vector field
Name2: A saddle point

Name: The index of \(\Gamma_s\) with respect to \(V\)
Definition: The number of revolution of \(V(z)\) as \(z\) traverses \(\Gamma(s)\)
Page: 456, 458
Chapter: 10II
Section: 1
Variables: \(V\) a vector field
\(\Gamma_s\) a simple loop
Notation: \(\mathscr I_V[\Gamma_s]\)
Definition2: \(\nu[V(\Gamma_s),0]\)

Question: \(\mathscr I_V(\Gamma_s)\) equals
Answer: -1
Page: 456
Chapter: 10II
Section: 1
Variables: \(V\) a vector field
\(\Gamma_s\) a simple loop

Name: The index of \(s\) with respect to \(V\)
Definition: The index of any simple loop around \(s\) with respect to \(V\)
Page: 456
Chapter: 10II
Section: 1
Variables: \(V\) a vector field
\(s\in\mathbb C\)
Notation: \(\mathscr I(s)\)
Example: 1, 1
Example2: -1
Example3: -2
Applied to: ,
Example4: 2
Applied to4: a dipole

Name: a dipole
Page: 457
Chapter: 10II
Section: 1
Context: \(\mathbb C\)

Name: a sink
Page: 457
Chapter: 10II
Section: 1
Context: \(\mathbb C\)

Name: a double crosspoint
Page: 457
Chapter: 10II
Section: 1
Context: \(\mathbb C\)

Name: A vortex
Page: 457
Chapter: 10II
Section: 1
Context: \(\mathbb C\)

Question: The index of the loop \(-\Gamma_s\) around \(s\) equals
Answer: the opposite of the index of \(\Gamma_s\) around \(s\)
Page: 458
Chapter: 10II
Section: 1
Variables: \(V\) a vector field
\(\Gamma_s\) a simple loop
\(s\in\mathbb C\)
Index: ii

Smaller0: \(s\) is non-singular
Greater0: \(\mathscr I(s)=0\)
Page: 458
Chapter: 10II
Section: 1
Variables: \(V\) a vector field
\(\Gamma_s\) a simple loop
\(s\in\mathbb C\)
Index: ii

Question: \(\mathscr I_{-V}[\Gamma_s]=\)
Answer: \(\mathscr I_{V}[\Gamma_s]\)
Page: 458
Chapter: 10II
Section: 1
Variables: \(V\) a vector field
\(\Gamma_s\) a simple loop
\(s\in\mathbb C\)
Index: ii

Question: the index of \(\mathscr I\) at \(s\) is
Answer: the opposite of the order of the pole at \(s\)
Page: 458
Chapter: 10II
Section: 1
Variables: \(V\) a vector field
\(\Gamma_s\) a simple loop
\(s\in\mathbb C\)
Index: ii
Assuming: \(s\) is a pole

Question: the index of \(\mathscr I\) at \(s\) is
Answer: the order of the root at \(s\)
Page: 458
Chapter: 10II
Section: 1
Variables: \(V\) a vector field
\(\Gamma_s\) a simple loop
\(s\in\mathbb C\)
Index: ii
Assuming: \(s\) is a root

Question: Difference between \(\mathscr I_V(\cdot)\) and \(\mathscr I_V[\cdot]\)
Answer: \(\mathscr I_V[\Gamma_s]\) takes as argument a curve and \(\mathscr I_V(s)\) a complex point
Page: 456
Chapter: 10II
Section: 1

Smaller0: \(\sum c_iz^i\) and \(\sum d_iz^i\) aggree on an infinite sequence converging to \(0\)
Greater0: \(\sum c_iz^i=\sum d_iz^i\)
Page: 71
Chapter: 2I
Section: 4
Variables: \((c_i)_i,(d_i)_i\) two sequences of complex
Smaller to greater: iff
Construction: \(n\) points determine a polynome of degree \(n\), let \(n\to\infty\)
Greatest: \((c_i)_i=(d_i)_i\)

Smaller0: \(a^1=a^2\)
Greater0: equals \(\sum_i (c^1_i+c^2_i)(z-a^1)^i\) and its radius of convergence is at least \(\min(D^1,D^2)\)
Page: 72
Chapter: 2I
Section: 5
Variables: \((c^1_i),(c^2_i)\) two sequences of complex
\(f^j(z)=\sum_i c^j_i(z-a^j)^i\) with radius of convergence \(D^j\) and center \(a^j\)
Prefix Greater: \(f^1(z)+f^2(z)\)

Smaller0: \(a^1=a^2\)
Greater0: equals \(\sum_i\left(\sum_{k=0}^i c^1_kc^2_{i-k}\right)(z-a^1)^i\) and its radius of convergence is at least \(\min(D^1,D^2)\)
Page: 72
Chapter: 2I
Section: 5
Variables: \((c^1_i),(c^2_i)\) two sequences of complex
\(f^j(z)=\sum_i c^j_i(z-a^j)^i\) with radius of convergence \(D^j\) and center \(a^j\)
Prefix Greater: \(f^1(z)f^2(z)\)