Gradiente, superficies equipotenciais e campo gradiente de funcións de tres variables $f(x,y,z)$
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"""
A función plot3d_vector_field é de Trevor Arnold
"""
def plot3d_vector_field(f,xrange,yrange,zrange,grid=[5,5,5],thick_arrows=False,**kwds):
"""
Return a 3-dimensional representation of a vector field contained
in the specified frame.
INPUT:
f -- a 3-tuple (or list) of functions or expressions
xrange, yrange, zrange -- 3-tuples of the form (var,var_min,var_max)
defining a box in which to display the plot
grid -- (default: [5,5,5]) a list specifying how many arrows
to draw in each dimension
thick_arrows -- (default: False) setting this to True may slow
things down a bit...
"""
from sage.ext.fast_eval import fast_float
from sage.plot.plot3d.shapes2 import frame3d
framepts = [(xrange[1],yrange[1],zrange[1]),(xrange[2],yrange[2],zrange[2])]
x = xrange[0]
y = yrange[0]
z = zrange[0]
xstep = (framepts[1][0]-framepts[0][0])/grid[0]
ystep = (framepts[1][1]-framepts[0][1])/grid[1]
zstep = (framepts[1][2]-framepts[0][2])/grid[2]
gridpoints = [(framepts[0][0] + xstep*(i+0.5),\
framepts[0][1] + ystep*(j+0.5), framepts[0][2] + zstep*(k+0.5))\
for i in range(grid[0]) for j in range(grid[1]) for k in range(grid[2])]
f_fast = fast_float(tuple(f),str(x),str(y),str(z))
f_vals = [vector((f_fast[0](*point),f_fast[1](*point),f_fast[2](*point))) for point in gridpoints]
fmax = max([val.norm() for val in f_vals])
theplot = line3d(framepts,opacity=0) # stop the bounding box from shrinking
theplot += text3d(str(x),(.5*(framepts[0][0]+framepts[1][0]),framepts[0][1],framepts[0][2])) +\
text3d(str(y),(framepts[1][0],.5*(framepts[0][1]+framepts[1][1]),framepts[0][2])) +\
text3d(str(z),(framepts[0][0],framepts[0][1],.5*(framepts[0][2]+framepts[1][2])))
if thick_arrows:
radius = lambda offset: 0.1*abs(offset)
else:
radius = lambda offset: None
for i in range(len(gridpoints)):
offset = 0.5/fmax*vector((f_vals[i][0]*xstep,f_vals[i][1]*ystep,f_vals[i][2]*zstep))
tail = vector(gridpoints[i]) - offset
head = vector(gridpoints[i]) + offset
theplot += line3d([tail,head],arrow_head=True,radius=radius(offset),**kwds)
return theplot
var('x,y,z')
@interact
def _(f =input_box(default=x*y+2*x*y^2+z^2),rango_x='(-2, 2)', rango_y='(-2,2)', rango_z='(-2,2)',
num_superficies=slider(1,4,1,2)):
xmin, xmax = sage_eval(rango_x); ymin, ymax = sage_eval(rango_y);zmin,zmax=sage_eval(rango_z);
html('
O vector gradiente nun punto xenérico é $%s$
' %latex([f.diff(x),f.diff(y),f.diff(z)]))
demo=plot3d_vector_field([f.diff(x),f.diff(y),f.diff(z)],(x,xmin,xmax),(y,ymin,ymax),(z,zmin,zmax))
for k in range(num_superficies):
demo += implicit_plot3d(f==k,(x,xmin,xmax),(y,ymin,ymax),(z,zmin,zmax),
opacity=(1-k/num_superficies),rgbcolor=(.5,.5*((k+1)/num_superficies),1))
show(demo)
///