# Triple Integral Interact for general w=f(x,y,z). Draws the nice region in cartesian,
# cylindrical, and spherical with differential terms in various orders. Note that if
# f(x,y,z) = 1, then this is simply the volume of the region R. If not, this has no nice pic.
def dlist(vs):
vs = vs.split()
return [" ".join("d%s" % v for v in a) for a in Arrangements(vs, len(vs))]
var('x,y,z')
@interact(layout={
"top": [
["top0", "top1", "top2"],
["int0", "int1", "int2", "f", "dV", "auto_update"],
["bot0", "bot1", "bot2"],
],
"bottom": [
["plot_axes", "plot_origin"],[ "compute", "opacity"],
],
})
def _(
top0=input_box(label="", default=3, width=4),
int0=text_control(r"""<div class="math">\color{red}{\int}</div>"""),
bot0=input_box(label="", default=1, width=4),
top1=input_box(label="", default=3, width=8),
int1=text_control(r"""<div class="math">\color{green}{\int}</div>"""),
bot1=input_box(label="", default=1, width=8),
top2=input_box(label="", default=3, width=12),
int2=text_control(r"""<div class="math">\color{blue}{\int}</div>"""),
bot2=input_box(label="", default=1, width=12),
f=input_box(label="", default="1", width=30),
dV=selector(flatten(map(dlist, ["x y z", "r theta z", "rho theta phi"])), label="", default="dz dy dx"),
opacity=slider(range(101), label="Opacity:", default=75),
plot_axes=checkbox(label=" Axes:", default=True),
plot_origin=checkbox(label=" Origin:", default=True),
compute=checkbox(label=" Try to compute:", default=False),
auto_update=False
):
opacity /= 100
vs = [d[1:] for d in dV.split()]
vs.reverse()
V = map(SR, vs)
v0, v1, v2 = (SR.var(v) for v in vs)
# While ln = log, symbolically they are treated differently
limits = [eval("[v%d, SR(str(bot%d).replace('ln', 'log')), SR(str(top%d).replace('ln', 'log'))]" % (i, i, i)) for i in range(3)]
f = SR(str(f).replace('ln', 'log'))
if compute:
integrals = []
differentials = []
for v, a, b in limits:
integrals.append(r"\int_{%s}^{%s} " % (latex(a), latex(b)))
differentials.insert(0, r"\, d %s" % latex(v))
try:
assume(a < v)
assume(v < b)
except ValueError:
pass
expression = [r"\["]
integrand = f
try:
for i in [3, 2, 1]:
expression.extend(integrals[:i])
expression.append(latex(integrand))
expression.extend(differentials[-i:])
expression.append("=")
v, a, b = limits[i - 1]
try:
integrand = integral(integrand, v, a, b)
except:
integrand = integral(integrand, v)
integrand = (integrand.subs({v: b}) - integrand.subs({v: a})).simplify_full()
# Don't add these at once in case of exceptions from n().
expression.extend([latex(integrand), r"\approx"])
expression.append(latex(integrand.n()))
except:
expression.append(r"\dots ?")
forget()
expression.append(r"\]")
pretty_print(html(" ".join(expression)))
if "x" in vs:
def to_xyz(*V):
return [V[vs.index(v)] for v in "xyz"]
elif "r" in vs:
def to_xyz(*V):
r, theta, z = [V[vs.index(v)] for v in ["r", "theta", "z"]]
return [r*cos(theta), r*sin(theta), z]
else:
def to_xyz(*V):
rho, phi, theta = [V[vs.index(v)] for v in ["rho", "phi", "theta"]]
return [rho*sin(phi)*cos(theta), rho*sin(phi)*sin(theta), rho*cos(phi)]
try:
u = SR.var("u")
v = SR.var("v")
p = Graphics()
V1 = (1-u) * limits[1][1] + u * limits[1][2]
for i in [1,2]:
V2 = limits[2][i].subs({v0: v0, v1: V1})
p += parametric_plot3d(to_xyz(v0, V1, V2), limits[0], (u, 0, 1), opacity=opacity, color="blue")
for i in [1,2]:
V1 = limits[1][i]
V2 = (1-u) * limits[2][1].subs({v1: V1}) + u * limits[2][2].subs({v1: V1})
p += parametric_plot3d(to_xyz(v0, V1, V2), limits[0], (u, 0, 1), opacity=opacity, color="green", boundary_style={"color": "green", "thickness": 5})
for i in [1,2]:
V0 = limits[0][i]
V1 = (1-u) * limits[1][1].subs({v0: V0}) + u * limits[1][2].subs({v0: V0})
V2 = (1-v) * limits[2][1].subs({v0: V0, v1: V1}) + v * limits[2][2].subs({v0: V0, v1: V1})
p += parametric_plot3d(to_xyz(V0, V1, V2), (u, 0, 1), (v, 0, 1), opacity=opacity, color="red", boundary_style={"color": "red", "thickness": 10})
if plot_origin:
p += point3d((0, 0, 0), size=25, color="yellow")
if plot_axes:
bb = p.bounding_box()
for i, v in enumerate("XYZ"):
start = min(bb[0][i], 0)
end = max(bb[1][i], 0)
length = end - start
p += line([[0]*i+[start-0.08*length]+[0]*(2-i), [0]*i+[end+0.08*length]+[0]*(2-i)], thickness=3, color="brown")
p += text3d(v, [0]*i+[end+0.1*length]+[0]*(2-i))
p.show(frame=False, aspect_ratio=1)
except ValueError:
html(r"""<span style="color:red">Cannot draw the plot, is the integral valid?..</span>""")
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Click to the left again to hide and once more to show the dynamic interactive window
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