CoCalc -- Vector Fields.ipynb

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Image: ubuntu2204

Kernel: SageMath 9.8

In [1]:

var('x y')
plot_vector_field((x,y), (x,-5,5), (y,-5,5))

In [2]:

# Stewart section 16.3, example 2, not a conservative field
plot_vector_field((x-y, x-2), (x, -10, 10), (y,-10,10))

In [3]:

# Stewart section 16.3, example 3, a conservative field
plot_vector_field((3+2*x*y, x^2-3*y^2), (x,-2,2), (y,-2,2))

In [4]:

plot_vector_field((cos(x), sin(y)), (x,-3,3), (y,-3,3))

In [5]:

plot_vector_field((y, (cos(x)-2)*sin(x)), (x,-pi,pi), (y,-pi,pi))

In [6]:

var('x y z')
plot_vector_field3d((x, y, z), (x,0,pi), (y,0,pi), (z,0,pi), viewer='threejs')

In [7]:

plot_vector_field3d((x*cos(z),-y*cos(z),sin(z)), (x,0,pi), (y,0,pi), (z,0,pi), viewer='threejs')

In [8]:

f(x,y) = x^2 * sin(2*y)
f.gradient()

(x, y) |--> (2*x*sin(2*y), 2*x^2*cos(2*y))

In [9]:

plot_vector_field(f.gradient(), (-3,3), (-3,3))

In [10]:

f(x,y,z) = z * exp(-x*y)
plot_vector_field3d(f.gradient(), (x,-1,1), (y,-1,1), (z,-1,1), viewer='threejs')

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
Cell In [10], line 2
      1 __tmp__=var("x,y,z"); f = symbolic_expression(z * exp(-x*y)).function(x,y,z)
----> 2 plot_vector_field3d(f.gradient(), (x,-Integer(1),Integer(1)), (y,-Integer(1),Integer(1)), (z,-Integer(1),Integer(1)), viewer='threejs')
File /ext/sage/9.8/src/sage/plot/plot3d/plot_field3d.py:148, in plot_vector_field3d(functions, xrange, yrange, zrange, plot_points, colors, center_arrows, **kwds)
    146     return G
    147 else:
--> 148     G = sum([plot(v, color=cm(v.norm()), **kwds).translate(p) for v, p in zip(scaled_vectors, points)])
    149     G._set_extra_kwds(kwds)
    150     return G
File /ext/sage/9.8/src/sage/plot/plot3d/plot_field3d.py:148, in <listcomp>(.0)
    146     return G
    147 else:
--> 148     G = sum([plot(v, color=cm(v.norm()), **kwds).translate(p) for v, p in zip(scaled_vectors, points)])
    149     G._set_extra_kwds(kwds)
    150     return G
File /ext/sage/9.8/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/matplotlib/colors.py:714, in Colormap.__call__(self, X, alpha, bytes)
    712 xa = np.array(X, copy=True)
    713 if mask_bad is None:
--> 714     mask_bad = np.isnan(xa)
    715 if not xa.dtype.isnative:
    716     xa = xa.byteswap().newbyteorder()  # Native byteorder is faster.
TypeError: ufunc 'isnan' not supported for the input types, and the inputs could not be safely coerced to any supported types according to the casting rule ''safe''

In [11]:

f(x,y) = x^2 + y^2
p1 = plot_vector_field(f.gradient(), (x,-2,2), (y,-2,2))
p2 = contour_plot(f, (x,-2,2), (y,-2,2), fill=False)
show(p1 + p2)

In [0]: