{
  "cells": [
     {"cell_type":"markdown","source":"<h1>Questions to be handed in on using Julia for parametric functions</h1>","metadata":{"internals":{"slide_type":"subslide","slide_helper":"subslide_end"},"slideshow":{"slide_type":"slide"},"slide_helper":"slide_end"}},
{"cell_type":"markdown","source":"<h3>Quick background</h3>","metadata":{"internals":{"slide_type":"subslide"},"slideshow":{"slide_type":"subslide"},"slide_helper":"slide_end"}},
{"cell_type":"markdown","source":"<p>Read the notes for details.</p>","metadata":{}},
{"cell_type":"markdown","source":"<p>The notes define a few useful functions, replicated here. (If the <code>MTH229</code> package is installed, this can <em>all</em> be replaced by <code>using MTH229</code>)</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/plain":["arrow! (generic function with 2 methods)"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["using LinearAlgebra # for norm, dot, and cross\nusing PyPlot        # for plotting\nusing SymPy         # for symbolic math\nusing Roots         # to find zeros numerically\nusing QuadGK        # to integerate numerically\nusing ForwardDiff   # for derivatives\n\nuvec(r) = r / norm(r) # unit vector\n\nD(r::Function, n=1) = n > 1 ? D(D(r),n-1) : t -> ForwardDiff.derivative(r, float(t))\nBase.adjoint(r::Function) = D(r)\n\nxs_ys(vs) = (A=hcat(vs...); Tuple([A[i,:] for i in eachindex(vs[1])]))\nxs_ys(v,vs...) = xs_ys([v, vs...])\nxs_ys(r::Function, a, b, n=100) = xs_ys(r.(range(a, stop=b, length=n)))\n\nfunction arrow!(plt::Plots.Plot, p, v; kwargs...)\n  if length(p) == 2\n     quiver!(plt, xs_ys([p])..., quiver=Tuple(xs_ys([v])); kwargs...)\n  elseif length(p) == 3\n    # 3d quiver needs support\n    # https://github.com/JuliaPlots/Plots.jl/issues/319#issue-159652535\n    # headless arrow instead\n    plot!(plt, xs_ys(p, p+v)...; kwargs...)\n\tend\nend\narrow!(p,v;kwargs...) = arrow!(Plots.current(), p, v; kwargs...)"],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<h2>Questions</h2>","metadata":{"internals":{"slide_type":"subslide","slide_helper":"subslide_end"},"slideshow":{"slide_type":"slide"},"slide_helper":"slide_end"}},
{"cell_type":"markdown","source":"<ul>\n<li><p>Let $u =\\langle 1,2,3 \\rangle$, $v=\\langle 1,1,1 \\rangle$, and $w = \\langle 1, -4, 3 \\rangle$. Use vector calculus with <code>julia</code> to find:</p>\n</li>\n</ul>","metadata":{}},
{"cell_type":"markdown","source":"<p>The sum $u + v$:</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<p>The value of $\\| v \\|$:</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<p>A unit vector $\\hat{u}$:</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<p>The angle between the vectors $u$ and $v$:</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<p>The volume of the parallelpiped formed by $u$, $v$, and $w$:</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<p>Verify that the cross product is anti-commutative by showing $u \\times v + v \\times u = 0$:</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<p>Verify that the cross product is not associative by computing both $(u \\times v) \\times w$ and $u \\times (v \\times w)$ and comparing the answers:</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<ul>\n<li><p>Let $p = \\langle 3,4,5 \\rangle$ and $v = \\langle 1,2,1 \\rangle$. Write a function, <code>r&#40;t&#41;</code>, that parameterizes the line in the direction of $v$ going through the point $p$.</p>\n</li>\n</ul>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<p>Find the distance between $r(0)$ and $r(3)$ using <code>julia</code>:</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<ul>\n<li><p>Let <code>r&#40;t&#41; &#61; &#91;sin&#40;t&#41;, 2cos&#40;t&#41;, 0&#93;</code>.</p>\n</li>\n</ul>","metadata":{}},
{"cell_type":"markdown","source":"<p>Make a plot of <code>r</code> over the interval $[0,2pi]$, what kind of shape is it?</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<p>Let <code>er</code> be the unit vector of <code>r</code>. Write this as a function of <code>t</code>:</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<p>Show that <code>r&#40;t&#41; × r&#39;&#40;t&#41;</code> is a constant in <code>t</code> by taking 3 different values of <code>t</code> and comparing:</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<p>Show that <code>r&#39;&#39;&#40;t&#41;</code> and <code>r&#40;t&#41;</code> are parallel by showing the for a 3 different values of <code>t</code> the cross product of the two vectors is basically 0 (up to numeric round off).</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<h2>CAS Questions</h2>","metadata":{"internals":{"slide_type":"subslide","slide_helper":"subslide_end"},"slideshow":{"slide_type":"slide"},"slide_helper":"slide_end"}},
{"cell_type":"markdown","source":"<p>Using <code>SymPy</code>, define a symbolic variable <code>t</code>:</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/plain":["(t,)"]},"metadata":{},"execution_count":1}],"cell_type":"code","source":["using SymPy\n@vars t real=true"],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<ul>\n<li><p>The Bernoulli spiral is parameterized by <code>r&#40;t&#41; &#61; exp&#40;t&#41; * &#91;cos&#40;4t&#41;, sin&#40;4t&#41;&#93;</code>. Plot this over $[-10, 3]$:</p>\n</li>\n</ul>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<p>Show the following symbolically for the Bernoulli spiral: the angle between the position vector $r(t)$ and the tangent vector $r'(t)$ is constant, that is it does not depend on $t$:</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<ul>\n<li><p>A spiraling curve is given by $r(t) = [t \\cdot \\cos(4t), t \\cdot \\sin(4t)]$.</p>\n</li>\n</ul>","metadata":{}},
{"cell_type":"markdown","source":"<p>Make a plot over $[0, 3\\pi]$:</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<p>Compute $\\| r'(t) \\|$ symbolically and assign its <em>simplified</em> output to <code>ds</code>.</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<p>Compute the arclength over $[0,3\\pi]$ via <code>integrate&#40;ds, t, 0, 3PI&#41;</code></p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<ul>\n<li><p>Let Viviani's curve be given by $r(t) = \\langle 1 + \\cos(t), \\sin(t), 2\\sin(t/2) \\rangle$.</p>\n</li>\n</ul>","metadata":{}},
{"cell_type":"markdown","source":"<p>Plot this over $[0, 2\\pi]$:</p>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<p>Compute the curvature of the curve at time $t$ and compare with</p>","metadata":{}},
{"cell_type":"markdown","source":"\n$$\n\\frac{\\sqrt{13 + 3\\cos(t)}}{(3 + \\cos(t))^{3/2}}\n$$\n","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1},
{"cell_type":"markdown","source":"<ul>\n<li><p>Let $r(t) = [\\sin(t), t, 0]$. Find the curvature for all $t$. Plot it. Where is it maximal? What part of the $\\sin(t)$ curve would this correspond to?</p>\n</li>\n</ul>","metadata":{}},
{"outputs":[{"output_type":"execute_result","data":{"text/latex":[""]},"metadata":{},"execution_count":1}],"cell_type":"code","source":[""],"metadata":{},"execution_count":1}
    ],
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