{
"Region of Integration Plotter" : {
"cat" : [ "Calculus" , "Multivariate" ],
"descr" : "Plotting regions of integration for iterated triple integrals in different coordinate systems.",
"code" : [
"# Limits of the left integral, first bottom, then top.",
"bot0d = \"1\"",
"top0d = \"3\"",
"# Limits of the middle integral.",
"bot1d = \"1\"",
"top1d = \"3\"",
"# Limits of the right integral.",
"bot2d = \"1\"",
"top2d = \"3\"",
"# Function to integrate.",
"fd = \"1\"",
"# Order of integration (must match one of the strings in the drop-down list).",
"dVd = \"dz dy dx\"",
"# Transparency control: 0 completely clear, 1 completely opaque.",
"opacity = 0.75",
"",
"#########################################",
"# Copyright (c) 2012 Andrey Novoseltsev #",
"#########################################",
"",
"html(r\"\"\"",
"<ul>",
"<li>Consider unchecking <b>Update plot</b> checkbox on the bottom while you are setting up the integral.</li>",
"<li>On the plot <span style=\"color:red\">red parts</span> correspond to points where limits of the <span style=\"color:red\">red integral</span> are reached, etc.</li>",
"<li>Try setting equal limits on the internal integral in cylindrical and spherical coordinates to see \"planes parallel to coordinate ones.\"</li>",
"</ul>",
"\"\"\")",
"",
"def dlist(vs):",
" vs = vs.split()",
" return [\" \".join(\"d%s\" % v for v in a) for a in Arrangements(vs, len(vs))]",
"",
"@interact(layout={",
"\"top\": [",
"[\"top0\", \"top1\", \"top2\"],",
"[\"int0\", \"int1\", \"int2\", \"f\", \"dV\"],",
"[\"bot0\", \"bot1\", \"bot2\"],",
"],",
"\"bottom\": [",
"[\"plot_axes\", \"plot_origin\", \"update_plot\", \"compute\"],",
"],",
"})",
"def _(",
" top0=input_box(label=\"\", default=top0d, type=str, width=5),",
" int0=text_control(r\"\"\"<div class=\"math\">\\color{red}{\\int}</div>\"\"\"),",
" bot0=input_box(label=\"\", default=bot0d, type=str, width=5),",
" top1=input_box(label=\"\", default=top1d, type=str, width=10),",
" int1=text_control(r\"\"\"<div class=\"math\">\\color{green}{\\int}</div>\"\"\"),",
" bot1=input_box(label=\"\", default=bot1d, type=str, width=10),",
" top2=input_box(label=\"\", default=top2d, type=str, width=15),",
" int2=text_control(r\"\"\"<div class=\"math\">\\color{blue}{\\int}</div>\"\"\"),",
" bot2=input_box(label=\"\", default=bot2d, type=str, width=15),",
" f=input_box(label=\"\", default=fd, type=str, width=40),",
" dV=selector(dlist(\"x y z\") + dlist(\"z r theta\") + dlist(\"rho phi theta\"), label=\"\", default=dVd),",
" plot_axes=checkbox(label=\"Axes:\", default=True),",
" plot_origin=checkbox(label=\" Origin:\", default=True),",
" update_plot=checkbox(label=\" Update plot:\", default=True),",
" compute=checkbox(label=\" Try to compute:\", default=False),",
" ):",
" vs = [d[1:] for d in dV.split()]",
" vs.reverse()",
" V = [SR(v) for v in vs]",
" v0, v1, v2 = (SR.var(v) for v in vs)",
" limits = [eval(\"[v%d, SR(bot%d), SR(top%d)]\" % (i, i, i)) for i in range(3)]",
" f(v0, v1, v2) = SR(f)",
"",
" def to_xyz(*V):",
" if \"x\" in vs:",
" return [V[vs.index(v)] for v in \"xyz\"]",
" elif \"r\" in vs:",
" r, theta, z = [V[vs.index(v)] for v in [\"r\", \"theta\", \"z\"]]",
" return [r*cos(theta), r*sin(theta), z]",
" else:",
" 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)]",
"",
" 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))",
" if update_plot:",
" p.show(frame=False, aspect_ratio=1)",
" if compute:",
" try:",
" I2 = integral(f, *(limits[2]))",
" I1 = integral(I2, *(limits[1]))",
" I0 = integral(I1, *(limits[0]))",
" html(r\"\\[ \\iiint_D %s\\, dV = %s \\approx %s \\]\" % tuple(map(latex, [f(v0, v1, v2), I0(0,0,0), I0.n()])))",
" except:",
" html(r\"\\[ \\iiint_D %s\\, dV = \\dots ? \\]\" % latex(f(v0, v1, v2)))"
]
}
}
