222 - Topic 19 - Vector Field Viewing 2D

604 days ago by Professor222

John Travis

Mississippi College

Vector Field Viewing.

In the interactive cell below, enter functions for each of the three components  of the 2D vector field $F(x,y) = <F_1,F_2>$.

%auto var('x,y,t') @interact def _(M=input_box(default=x-sin(y),label='M'), N=input_box(default=-y*cos(x),label='N'), range_x = input_grid(1, 2, default=[[-2,2]]), range_y = input_grid(1, 2, default=[[-2,2]]), more_pts=checkbox(default=False,label='More Vectors?'), even_more_pts=checkbox(default=False,label='Lots More Vectors?'), bazillion=checkbox(default=False,label='Way More Vectors?')): xmin = range_x[0][0] xmax = range_x[0][1] ymin = range_y[0][0] ymax = range_y[0][1] field = (M,N) if bazillion: density_vectors = 50 elif even_more_pts: density_vectors = 30 elif more_pts: density_vectors = 20 else: density_vectors = 10 H=plot_vector_field(field, (x,xmin,xmax), (y,ymin,ymax),plot_points=density_vectors,color='green') show(H) 
       
range_x 
range_y 
More Vectors? 
Lots More Vectors? 
Way More Vectors? 

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Now, instead let's create a conservative vector field by first providing a potential function and then letting the system create the corresponding field.

%auto var('x,y,t') @interact def _(f = input_box(default=x^3*y^3,label = "Potential Function ="), range_x = input_grid(1, 2, default=[[-2,2]]), range_y = input_grid(1, 2, default=[[-2,2]]), more_pts=checkbox(default=False,label='More Vectors?'), even_more_pts=checkbox(default=False,label='Lots More Vectors?'), bazillion=checkbox(default=False,label='Way More Vectors?')): xmin = range_x[0][0] xmax = range_x[0][1] ymin = range_y[0][0] ymax = range_y[0][1] M=derivative(f,x) N=derivative(f,y) pretty_print(html("Vector Field = $< %s$"%(str(latex(M)))+" , "+"$ %s$"%(str(latex(N))))) field = (M,N) if bazillion: density_vectors = 50 elif even_more_pts: density_vectors = 30 elif more_pts: density_vectors = 20 else: density_vectors = 10 H=plot_vector_field(field, (x,xmin,xmax), (y,ymin,ymax),plot_points=density_vectors,color='green') show(H) 
       
Potential Function = 
range_x 
range_y 
More Vectors? 
Lots More Vectors? 
Way More Vectors? 

Click to the left again to hide and once more to show the dynamic interactive window

%auto var('x,y,t') @interact def _(A=input_box(default=2*x*y*e^(x^2*y),label='$F_1(x,y) = $'), B=input_box(default=x^2*e^(x^2*y),label='$F_2(x,y) = $'), more_pts=checkbox(default=False,label='More Vectors?'), even_more_pts=checkbox(default=False,label='Lots More Vectors?'), bazillion=checkbox(default=False,label='Way More Vectors?')): field = (A,B) f = e^(x^2*y)+2 gradf = (derivative(f,x),derivative(f,y)) if bazillion: density_vectors = 50 elif even_more_pts: density_vectors = 30 elif more_pts: density_vectors = 20 else: density_vectors = 10 H=plot_vector_field(field, (x,-1,1), (y,-1,1),plot_points=density_vectors,color='green') # H+=plot_vector_field(gradf, (x,-1,1), (y,-1,1),plot_points=density_vectors,color='pink') show(H) 
       
$F_1(x,y) = $ 
$F_2(x,y) = $ 
More Vectors? 
Lots More Vectors? 
Way More Vectors? 

Click to the left again to hide and once more to show the dynamic interactive window

%auto from sage.plot.primitive import GraphicPrimitive from sage.misc.decorators import options from sage.misc.misc import xsrange # Below is the base class that is used to make 'field plots'. # Its implementation is motivated by 'PlotField'. # Currently it is used to make the functions 'plot_vector_field' # and 'plot_slope_field'. # TODO: use this to make these functions: # 'plot_gradient_field' and 'plot_hamiltonian_field' class PlotField_local(GraphicPrimitive): """ Primitive class that initializes the PlotField graphics type """ def __init__(self, xpos_array, ypos_array, xvec_array, yvec_array, options): """ Create the graphics primitive PlotField. This sets options and the array to be plotted as attributes. EXAMPLES:: sage: x,y = var('x,y') sage: R=plot_slope_field(x+y,(x,0,1),(y,0,1),plot_points=2) sage: r=R[0] sage: r.options()['headaxislength'] 0 sage: r.xpos_array [0.0, 0.0, 1.0, 1.0] sage: r.yvec_array masked_array(data = [0.0 0.707106781187 0.707106781187 0.894427191], mask = [False False False False], fill_value = 1e+20) TESTS: We test dumping and loading a plot:: sage: x,y = var('x,y') sage: P = plot_vector_field((sin(x), cos(y)), (x,-3,3), (y,-3,3)) sage: Q = loads(dumps(P)) """ self.xpos_array = xpos_array self.ypos_array = ypos_array self.xvec_array = xvec_array self.yvec_array = yvec_array GraphicPrimitive.__init__(self, options) def get_minmax_data(self): """ Returns a dictionary with the bounding box data. EXAMPLES:: sage: x,y = var('x,y') sage: d = plot_vector_field((.01*x,x+y), (x,10,20), (y,10,20))[0].get_minmax_data() sage: d['xmin'] 10.0 sage: d['ymin'] 10.0 """ from sage.plot.plot import minmax_data return minmax_data(self.xpos_array, self.ypos_array, dict=True) def _allowed_options(self): """ Returns a dictionary with allowed options for PlotField. EXAMPLES:: sage: x,y = var('x,y') sage: P=plot_vector_field((sin(x), cos(y)), (x,-3,3), (y,-3,3)) sage: d=P[0]._allowed_options() sage: d['pivot'] 'Where the arrow should be placed in relation to the point (tail, middle, tip)' """ return {'plot_points':'How many points to use for plotting precision', 'pivot': 'Where the arrow should be placed in relation to the point (tail, middle, tip)', 'headwidth': 'Head width as multiple of shaft width, default is 3', 'headlength': 'head length as multiple of shaft width, default is 5', 'headaxislength': 'head length at shaft intersection, default is 4.5', 'zorder':'The layer level in which to draw', 'color':'The color of the arrows'} def _repr_(self): """ String representation of PlotField graphics primitive. EXAMPLES:: sage: x,y = var('x,y') sage: P=plot_vector_field((sin(x), cos(y)), (x,-3,3), (y,-3,3)) sage: P[0] PlotField defined by a 20 x 20 vector grid """ return "PlotField defined by a %s x %s vector grid"%(self.options()['plot_points'], self.options()['plot_points']) def _render_on_subplot(self, subplot): """ TESTS:: sage: x,y = var('x,y') sage: P=plot_vector_field((sin(x), cos(y)), (x,-3,3), (y,-3,3)) """ options = self.options() quiver_options = options.copy() quiver_options.pop('plot_points') subplot.quiver(self.xpos_array, self.ypos_array, self.xvec_array, self.yvec_array, angles='xy', **quiver_options) @options(plot_points=20,frame=True) def plot_vector_field_unscaled((f, g), xrange, yrange, **options): r""" ``plot_vector_field`` takes two functions of two variables xvar and yvar (for instance, if the variables are `x` and `y`, take `(f(x,y), g(x,y))`) and plots vector arrows of the function over the specified ranges, with xrange being of xvar between xmin and xmax, and yrange similarly (see below). ``plot_vector_field((f, g), (xvar, xmin, xmax), (yvar, ymin, ymax))`` EXAMPLES: Plot some vector fields involving sin and cos:: sage: x,y = var('x y') sage: plot_vector_field((sin(x), cos(y)), (x,-3,3), (y,-3,3)) :: sage: plot_vector_field(( y, (cos(x)-2)*sin(x)), (x,-pi,pi), (y,-pi,pi)) Plot a gradient field:: sage: u,v = var('u v') sage: f = exp(-(u^2+v^2)) sage: plot_vector_field(f.gradient(), (u,-2,2), (v,-2,2), color='blue') Plot two orthogonal vector fields:: sage: x,y = var('x,y') sage: a=plot_vector_field((x,y), (x,-3,3),(y,-3,3),color='blue') sage: b=plot_vector_field((y,-x),(x,-3,3),(y,-3,3),color='red') sage: show(a+b) We ignore function values that are infinite or NaN:: sage: x,y = var('x,y') sage: plot_vector_field( (-x/sqrt(x^2+y^2), -y/sqrt(x^2+y^2)), (x, -10, 10), (y, -10, 10)) :: sage: x,y = var('x,y') sage: plot_vector_field( (-x/sqrt(x+y), -y/sqrt(x+y)), (x, -10, 10), (y, -10, 10)) Extra options will get passed on to show(), as long as they are valid:: sage: plot_vector_field((x, y), (x, -2, 2), (y, -2, 2), xmax=10) sage: plot_vector_field((x, y), (x, -2, 2), (y, -2, 2)).show(xmax=10) # These are equivalent """ from sage.plot.all import Graphics from sage.plot.misc import setup_for_eval_on_grid z, ranges = setup_for_eval_on_grid([f,g], [xrange, yrange], options['plot_points']) f,g = z xpos_array, ypos_array, xvec_array, yvec_array = [],[],[],[] for x in xsrange(*ranges[0], include_endpoint=True): for y in xsrange(*ranges[1], include_endpoint=True): xpos_array.append(x) ypos_array.append(y) xvec_array.append(f(x,y)) yvec_array.append(g(x,y)) import numpy xvec_array = numpy.ma.masked_invalid(numpy.array(xvec_array, dtype=float)) yvec_array = numpy.ma.masked_invalid(numpy.array(yvec_array, dtype=float)) g = Graphics() g._set_extra_kwds(Graphics._extract_kwds_for_show(options)) g.add_primitive(PlotField_local(xpos_array, ypos_array, xvec_array, yvec_array, options)) return g 
       
%auto var('x,y,t') @interact def _(A=input_box(default=x-sin(y),label='$F_1(x,y) = $'), B=input_box(default=-y*cos(x),label='$F_2(x,y) = $'), more_pts=checkbox(default=False,label='More Vectors?'), even_more_pts=checkbox(default=False,label='Lots More Vectors?'), bazillion=checkbox(default=False,label='Way More Vectors?')): field = (A,B) if bazillion: density_vectors = 50 elif even_more_pts: density_vectors = 30 elif more_pts: density_vectors = 20 else: density_vectors = 10 H=plot_vector_field_unscaled(field, (x,0,2*pi), (y,0,2*pi),plot_points=density_vectors,color='green') show(H) 
       
$F_1(x,y) = $ 
$F_2(x,y) = $ 
More Vectors? 
Lots More Vectors? 
Way More Vectors? 

Click to the left again to hide and once more to show the dynamic interactive window