Fractals/Mathematics/Vector field

Vector field

Here mainly numerical methods for time independent 2D vector fields are described.

Algorithm

• plane (parameter plane or dynamic plane)
• scalar function
• vector function
• create scalar field using scalar function ( potential)
• create vector field from scalar field using vector function ( gradient of the potential)
• compute:
• filed lines ( stream lines )
• contour lines ( equipotential lines )
• map whole field using
• Line Integral Convolution (LIC)

Field line integration method

rk4

Fourth-order Runge-Kutta (RK4) in case of 2D time independent vector field

$F$ is a vector function that for each point p

p = (x, y)

in a domain assigns a vector v

$v=F(p)=F(x,y)=(F_{1}(x,y),F_{2}(x,y))$ where each of the functions $F_{i}$ is a scalar function:

$F_{i}:\mathbb {R} ^{2}\to \mathbb {R}$ A field line is a line that is everywhere tangent to a given vector field.

Let r(s) be a field line given by a system of ordinary differential equations, which written on vector form is:

${\frac {dr}{ds}}=F(r(s))$ where:

• s representing the arc length along the field line
• $r(0)=r_{0}$ is a seed point

Given a seed point $r_{0}$ on the field line, the update rule ( RK4) to find the next point $r_{i}$ along the field line is

$r_{i+1}=r_{i}+{\frac {h(k_{1}+2k_{2}+2k_{3}+k_{4})}{6}}$ where:

• h is the step size
• k are the intermediate vectors:

${\begin{array}{lcl}k_{1}=F(r_{i})\\k_{2}=F(r_{i}+{\frac {h}{2}}k_{1})\\k_{3}=F(r_{i}+{\frac {h}{2}}k_{2})\\k_{4}=F(r_{i}+hk_{3})\end{array}}$ Visualisation of vector field

Plot types (Visualization Techniques for Flow Data) : 

• Glyphs = Icons or signs for visualizing vector fields
• simplest glyph = Line segment (hedgehog plots)
• arrow plot = quiver plot = Hedgehogs (global arrow plots)
• Characteristic Lines 
• stremlines = curve everywhere tangential to the instantaneous vector (velocity) field (time independent vector field). For time independent vector field streaklines = Path lines = streak lines 
• texture (line integral convolution = LIC)
• Topological skeleton 
• fixed point extraction ( Jacobian)

Dictionary

• vector function is a function that gives vector as an output
• field : space (plane, sphere, ... )
• field line is a line that is everywhere tangent to a given vector field
• scalar/ vector / tensor:
• Scalars are real numbers used in linear algebra. Scalar is a tensor of zero order
• Vector is a tensor of first order. Vector is an extension of scalar
• tensor is an extension of vector

Vector

Forms of 2D vector:

• [z1] ( only one complex number when first point is known , for example z0 is origin
• [z0, z1] = two complex numbers
• 4 scalars ( real numbers)
• [x, y, dx , dy]
• [x0, y0, x1, y1]
• [x, y, angle, magnitude]
• 2 scalars : [x1, y1] for second complex number when first point is known , for example z0 is origin

• LIC