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quarto/_freeze/differentiable_vector_calculus/plots_plotting/execute-results/html.json
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"markdown": "# 2D and 3D plots in Julia with Plots\n\n\n\nThis section uses these add-on packages:\n\n``` {.julia .cell-code}\nusing CalculusWithJulia\nusing Plots\nimport Contour: contours, levels, level, lines, coordinates\nusing LinearAlgebra\nusing ForwardDiff\n```\n\n\n\n\n---\n\n\nThis covers plotting the typical 2D and 3D plots in Julia with the `Plots` package.\n\n\nWe will make use of some helper functions that will simplify plotting provided by the `CalculusWithJulia` package. As well, we will need to manipulate contours directly, so pull in the `Contours` package, using `import` to avoid name collisions and explicitly listing the methods we will use.\n\n\n## Parametrically described curves in space\n\n\nLet $r(t)$ be a vector-valued function with values in $R^d$, $d$ being $2$ or $3$. A familiar example is the equation for a line that travels in the direction of $\\vec{v}$ and goes through the point $P$: $r(t) = P + t \\cdot \\vec{v}$. A *parametric plot* over $[a,b]$ is the collection of all points $r(t)$ for $a \\leq t \\leq b$.\n\n\nIn `Plots`, parameterized curves can be plotted through two interfaces, here illustrated for $d=2$: `plot(f1, f2, a, b)` or `plot(xs, ys)`. The former is convenient for some cases, but typically we will have a function `r(t)` which is vector-valued, as opposed to a vector of functions. As such, we only discuss the latter.\n\n\nAn example helps illustrate. Suppose $r(t) = \\langle \\sin(t), 2\\cos(t) \\rangle$ and the goal is to plot the full ellipse by plotting over $0 \\leq t \\leq 2\\pi$. As with plotting of curves, the goal would be to take many points between `a` and `b` and from there generate the $x$ values and $y$ values.\n\n\nLet's see this with 5 points, the first and last being identical due to the curve:\n\n::: {.cell execution_count=4}\n``` {.julia .cell-code}\nr₂(t) = [sin(t), 2cos(t)]\nts = range(0, stop=2pi, length=5)\n```\n\n::: {.cell-output .cell-output-display execution_count=5}\n```\n0.0:1.5707963267948966:6.283185307179586\n```\n:::\n:::\n\n\nThen we can create the $5$ points easily through broadcasting:\n\n::: {.cell execution_count=5}\n``` {.julia .cell-code}\nvs = r₂.(ts)\n```\n\n::: {.cell-output .cell-output-display execution_count=6}\n```\n5-element Vector{Vector{Float64}}:\n [0.0, 2.0]\n [1.0, 1.2246467991473532e-16]\n [1.2246467991473532e-16, -2.0]\n [-1.0, -3.6739403974420594e-16]\n [-2.4492935982947064e-16, 2.0]\n```\n:::\n:::\n\n\nThis returns a vector of points (stored as vectors). The plotting function wants two collections: the set of $x$ values for the points and the set of $y$ values. The data needs to be generated differently or reshaped. The function `unzip` above takes data in this style and returns the desired format, returning a tuple with the $x$ values and $y$ values pulled out:\n\n::: {.cell execution_count=6}\n``` {.julia .cell-code}\nunzip(vs)\n```\n\n::: {.cell-output .cell-output-display execution_count=7}\n```\n([0.0, 1.0, 1.2246467991473532e-16, -1.0, -2.4492935982947064e-16], [2.0, 1.2246467991473532e-16, -2.0, -3.6739403974420594e-16, 2.0])\n```\n:::\n:::\n\n\nTo plot this, we \"splat\" the tuple so that `plot` gets the arguments separately:\n\n::: {.cell execution_count=7}\n``` {.julia .cell-code}\nplot(unzip(vs)...)\n```\n\n::: {.cell-output .cell-output-display execution_count=8}\n![](plots_plotting_files/figure-html/cell-8-output-1.svg){}\n:::\n:::\n\n\nThis basic plot is lacking, of course, as there are not enough points. Using more initially is a remedy.\n\n::: {.cell hold='true' execution_count=8}\n``` {.julia .cell-code}\nts = range(0, 2pi, length=100)\nplot(unzip(r₂.(ts))...)\n```\n\n::: {.cell-output .cell-output-display execution_count=9}\n![](plots_plotting_files/figure-html/cell-9-output-1.svg){}\n:::\n:::\n\n\nAs a convenience, `CalculusWithJulia` provides `plot_parametric` to produce this plot. The interval is specified with the `a..b` notation of `IntervalSets` (which is available when the `CalculusWithJulia` package is loaded), the points to plot are adaptively chosen:\n\n::: {.cell execution_count=9}\n``` {.julia .cell-code}\nplot_parametric(0..2pi, r₂) # interval first\n```\n\n::: {.cell-output .cell-output-display execution_count=10}\n![](plots_plotting_files/figure-html/cell-10-output-1.svg){}\n:::\n:::\n\n\n### Plotting a space curve in 3 dimensions\n\n\nA parametrically described curve in 3D is similarly created. For example, a helix is described mathematically by $r(t) = \\langle \\sin(t), \\cos(t), t \\rangle$. Here we graph two turns:\n\n::: {.cell execution_count=10}\n``` {.julia .cell-code}\nr₃(t) = [sin(t), cos(t), t]\nplot_parametric(0..4pi, r₃)\n```\n\n::: {.cell-output .cell-output-display execution_count=11}\n![](plots_plotting_files/figure-html/cell-11-output-1.svg){}\n:::\n:::\n\n\n### Adding a vector\n\n\nThe tangent vector indicates the instantaneous direction one would travel were they walking along the space curve. We can add a tangent vector to the graphLine truncated
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"markdown": "# Polar Coordinates and Curves\n\n\n\nThis section uses these add-on packages:\n\n``` {.julia .cell-code}\nusing CalculusWithJulia\nusing Plots\nusing SymPy\nusing Roots\nusing QuadGK\n```\n\n\n\n\n---\n\n\nThe description of the $x$-$y$ plane via Cartesian coordinates is not the only possible way, though one that is most familiar. Here we discuss a different means. Instead of talking about over and up from an origin, we focus on a direction and a distance from the origin.\n\n\n## Definition of polar coordinates\n\n\nPolar coordinates parameterize the plane though an angle $\\theta$ made from the positive ray of the $x$ axis and a radius $r$.\n\n::: {.cell hold='true' execution_count=4}\n\n::: {.cell-output .cell-output-display execution_count=5}\n![](polar_coordinates_files/figure-html/cell-5-output-1.svg){}\n:::\n:::\n\n\nTo recover the Cartesian coordinates from the pair $(r,\\theta)$, we have these formulas from [right](http://en.wikipedia.org/wiki/Polar_coordinate_system#Converting_between_polar_and_Cartesian_coordinates) triangle geometry:\n\n\n\n$$\nx = r \\cos(\\theta),~ y = r \\sin(\\theta).\n$$\n\n\nEach point $(x,y)$ corresponds to several possible values of $(r,\\theta)$, as any integer multiple of $2\\pi$ added to $\\theta$ will describe the same point. Except for the origin, there is only one pair when we restrict to $r > 0$ and $0 \\leq \\theta < 2\\pi$.\n\n\nFor values in the first and fourth quadrants (the range of $\\tan^{-1}(x)$), we have:\n\n\n\n$$\nr = \\sqrt{x^2 + y^2},~ \\theta=\\tan^{-1}(y/x).\n$$\n\n\nFor the other two quadrants, the signs of $y$ and $x$ must be considered. This is done with the function `atan` when two arguments are used.\n\n\nFor example, $(-3, 4)$ would have polar coordinates:\n\n::: {.cell execution_count=5}\n``` {.julia .cell-code}\nx,y = -3, 4\nrad, theta = sqrt(x^2 + y^2), atan(y, x)\n```\n\n::: {.cell-output .cell-output-display execution_count=6}\n```\n(5.0, 2.214297435588181)\n```\n:::\n:::\n\n\nAnd reversing\n\n::: {.cell execution_count=6}\n``` {.julia .cell-code}\nrad*cos(theta), rad*sin(theta)\n```\n\n::: {.cell-output .cell-output-display execution_count=7}\n```\n(-2.999999999999999, 4.000000000000001)\n```\n:::\n:::\n\n\nThis figure illustrates:\n\n::: {.cell hold='true' execution_count=7}\n\n::: {.cell-output .cell-output-display execution_count=8}\n![](polar_coordinates_files/figure-html/cell-8-output-1.svg){}\n:::\n:::\n\n\nThe case where $r < 0$ is handled by going $180$ degrees in the opposite direction, in other words the point $(r, \\theta)$ can be described as well by $(-r,\\theta+\\pi)$.\n\n\n## Parameterizing curves using polar coordinates\n\n\nIf $r=r(\\theta)$, then the parameterized curve $(r(\\theta), \\theta)$ is just the set of points generated as $\\theta$ ranges over some set of values. There are many examples of parameterized curves that simplify what might be a complicated presentation in Cartesian coordinates.\n\n\nFor example, a circle has the form $x^2 + y^2 = R^2$. Whereas parameterized by polar coordinates it is just $r(\\theta) = R$, or a constant function.\n\n\nThe circle centered at $(r_0, \\gamma)$ (in polar coordinates) with radius $R$ has a more involved description in polar coordinates:\n\n\n\n$$\nr(\\theta) = r_0 \\cos(\\theta - \\gamma) + \\sqrt{R^2 - r_0^2\\sin^2(\\theta - \\gamma)}.\n$$\n\n\nThe case where $r_0 > R$ will not be defined for all values of $\\theta$, only when $|\\sin(\\theta-\\gamma)| \\leq R/r_0$.\n\n\n#### Examples\n\n\nThe `Plots.jl` package provides a means to visualize polar plots through `plot(thetas, rs, proj=:polar)`. For example, to plot a circe with $r_0=1/2$ and $\\gamma=\\pi/6$ we would have:\n\n::: {.cell hold='true' execution_count=8}\n``` {.julia .cell-code}\nR, r0, gamma = 1, 1/2, pi/6\nr(theta) = r0 * cos(theta-gamma) + sqrt(R^2 - r0^2*sin(theta-gamma)^2)\nts = range(0, 2pi, length=100)\nrs = r.(ts)\nplot(ts, rs, proj=:polar, legend=false)\n```\n\n::: {.cell-output .cell-output-display execution_count=9}\n![](polar_coordinates_files/figure-html/cell-9-output-1.svg){}\n:::\n:::\n\n\nTo avoid having to create values for $\\theta$ and values for $r$, the `CalculusWithJulia` package provides a helper function, `plot_polar`. To distinguish it from other functions provided by `Plots`, the calling pattern is different. It specifies an interval to plot over by `a..b` and puts that first (this notation for closed intervals is from `IntervalSets`), followed by `r`. 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"markdown": "# Scalar functions\n\n\n\nThis section uses these add-on packages:\n\n``` {.julia .cell-code}\nusing CalculusWithJulia\nusing Plots\nusing ForwardDiff\nusing SymPy\nusing Roots\nusing QuadGK\nusing JSON\n```\n\n\nAlso, these methods from the `Contour` package:\n\n``` {.julia .cell-code}\nimport Contour: contours, levels, level, lines, coordinates\n```\n\n\n\n\n---\n\n\nConsider a function $f: R^n \\rightarrow R$. It has multiple arguments for its input (an $x_1, x_2, \\dots, x_n$) and only one, *scalar*, value for an output. Some simple examples might be:\n\n\n\n$$\n\\begin{align}\nf(x,y) &= x^2 + y^2\\\\\ng(x,y) &= x \\cdot y\\\\\nh(x,y) &= \\sin(x) \\cdot \\sin(y)\n\\end{align}\n$$\n\n\nFor two examples from real life consider the elevation Point Query Service (of the [USGS](https://nationalmap.gov/epqs/)) returns the elevation in international feet or meters for a specific latitude/longitude within the United States. The longitude can be associated to an $x$ coordinate, the latitude to a $y$ coordinate, and the elevation a $z$ coordinate, and as long as the region is small enough, the $x$-$y$ coordinates can be thought to lie on a plane. (A flat earth assumption.)\n\n\nSimilarly, a weather map, say of the United States, may show the maximum predicted temperature for a given day. This describes a function that take a position ($x$, $y$) and returns a predicted temperature ($z$).\n\n\nMathematically, we may describe the values $(x,y)$ in terms of a point, $P=(x,y)$ or a vector $\\vec{v} = \\langle x, y \\rangle$ using the identification of a point with a vector. As convenient, we may write any of $f(x,y)$, $f(P)$, or $f(\\vec{v})$ to describe the evaluation of $f$ at the value $x$ and $y$\n\n\n---\n\nReturning to the task at hand, in `Julia`, defining a scalar function is straightforward, the syntax following mathematical notation:\n\n::: {.cell execution_count=5}\n``` {.julia .cell-code}\nf(x,y) = x^2 + y^2\ng(x,y) = x * y\nh(x,y) = sin(x) * sin(y)\n```\n\n::: {.cell-output .cell-output-display execution_count=6}\n```\nh (generic function with 1 method)\n```\n:::\n:::\n\n\nTo call a scalar function for specific values of $x$ and $y$ is also similar to the mathematical case:\n\n::: {.cell execution_count=6}\n``` {.julia .cell-code}\nf(1,2), g(2, 3), h(3,4)\n```\n\n::: {.cell-output .cell-output-display execution_count=7}\n```\n(5, 6, -0.10679997423758245)\n```\n:::\n:::\n\n\nIt may be advantageous to have the values as a vector or a point, as in `v=[x,y]`. Splatting can be used to turn a vector or tuple into two arguments:\n\n::: {.cell execution_count=7}\n``` {.julia .cell-code}\nv = [1,2]\nf(v...)\n```\n\n::: {.cell-output .cell-output-display execution_count=8}\n```\n5\n```\n:::\n:::\n\n\nAlternatively, the function may be defined using a vector argument:\n\n::: {.cell execution_count=8}\n``` {.julia .cell-code}\nf(v) = v[1]^2 + v[2]^2\n```\n\n::: {.cell-output .cell-output-display execution_count=9}\n```\nf (generic function with 2 methods)\n```\n:::\n:::\n\n\nA style required for other packages within the `Julia` ecosystem, as there are many advantages to passing containers of values: they can have arbitrary length, they can be modified inside a function, the functions can be more generic, etc.\n\n\nMore verbosely, but avoiding index notation, we can use multiline functions:\n\n::: {.cell execution_count=9}\n``` {.julia .cell-code}\nfunction g(v)\n x, y = v\n x * y\nend\n```\n\n::: {.cell-output .cell-output-display execution_count=10}\n```\ng (generic function with 2 methods)\n```\n:::\n:::\n\n\nThen we have\n\n::: {.cell execution_count=10}\n``` {.julia .cell-code}\nf(v), g([2,3])\n```\n\n::: {.cell-output .cell-output-display execution_count=11}\n```\n(5, 6)\n```\n:::\n:::\n\n\n---\n\n\nMore elegantly, perhaps – and the approach we will use in this section – is to mirror the mathematical notation through multiple dispatch. If we define `j` for multiple variables, say with:\n\n::: {.cell execution_count=11}\n``` {.julia .cell-code}\nj(x,y) = x^2 - 2x*y^2\n```\n\n::: {.cell-output .cell-output-display execution_count=12}\n```\nj (generic function with 1 method)\n```\n:::\n:::\n\n\nThe we can define an alternative method with just a single variable and use splatting to turn it into multiple variables:\n\n::: {.cell execution_count=12}\n``` {.julia .cell-code}\nj(v) = j(v...)\n```\n\n::: {.cell-output .cell-output-display execution_count=13}\n```\nj (generic function with 2 methods)\n```\n:::\n:::\n\n\nThe we can call `j` with a vector or point:\n\n::: {.cell execution_count=13}\n``` {.julia .cell-code}\nj([1,2])\n```\n\n::: {.cell-output .cell-output-display execution_count=14}\n```\n-7\n```\n:::\n:::\n\n\nor by passing in the individual components:\n\n::: {.cell execution_count=14}\n``` {.julia .cell-code}\nj(1,2)\n```\n\n::: {.cell-output .cell-output-display execution_count=15}\n```\n-7\n```\n:::\n:::\n\n\n---\n\nFollowing a calculus perspective, we take up the questionLine truncated
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"markdown": "# Applications with scalar functions\n\n\n\nThis section uses these add-on packages:\n\n``` {.julia .cell-code}\nusing CalculusWithJulia\nusing Plots\nusing SymPy\nusing Roots\n```\n\n\nAnd the following from the `Contour` package:\n\n``` {.julia .cell-code}\nimport Contour: contours, levels, level, lines, coordinates\n```\n\n\n\n\nThis section presents different applications of scalar functions.\n\n\n## Tangent planes, linearization\n\n\nConsider the case $f:R^2 \\rightarrow R$. We visualize $z=f(x,y)$ through a surface. At a point $(a, b)$, this surface, if $f$ is sufficiently smooth, can be approximated by a flat area, or a plane. For example, the Northern hemisphere of the earth, might be modeled simplistically by $z = \\sqrt{R^2 - (x^2 + y^2)}$ for some $R$ and with the origin at the earth's core. The ancient view of a \"flat earth,\" can be more generously seen as identifying this tangent plane with the sphere. More apt for current times, is the use of GPS coordinates to describe location. The difference between any two coordinates is technically a distance on a curved, nearly spherical, surface. But if the two points are reasonably closes (miles, not tens of miles) and accuracy isn't of utmost importance (i.e., not used for self-driving cars), then the distance can be found from the Euclidean distance formula, $\\sqrt{(\\Delta\\text{latitude})^2 + \\Delta\\text{longitude})^2}$. That is, as if the points were on a plane, not a curved surface.\n\n\nFor the univariate case, the tangent line has many different uses. Here we see the tangent plane also does.\n\n\n### Equation of the tangent plane\n\n\nThe partial derivatives have the geometric view of being the derivative of the univariate functions $f(\\vec\\gamma_x(t))$ and $f(\\vec\\gamma_y(t))$, where $\\vec\\gamma_x$ moves just parallel to the $x$ axis (e.g. $\\langle t + a, b\\rangle$). and $\\vec\\gamma_y$ moves just parallel to the $y$ axis. The partial derivatives then are slopes of tangent lines to each curve. The tangent plane, should it exist, should match both slopes at a given point. With this observation, we can identify it.\n\n\nConsider $f(\\vec\\gamma_x)$ at a point $(a,b)$. The path has a tangent vector, which has \"slope\" $\\frac{\\partial f}{\\partial x}$. and in the direction of the $x$ axis, but not the $y$ axis, as does this vector: $\\langle 1, 0, \\frac{\\partial f}{\\partial x} \\rangle$. Similarly, this vector $\\langle 0, 1, \\frac{\\partial f}{\\partial y} \\rangle$ describes the tangent line to $f(\\vec\\gamma_y)$ a the point.\n\n\nThese two vectors will lie in the plane. The normal vector is found by their cross product:\n\n::: {.cell execution_count=5}\n``` {.julia .cell-code}\n@syms f_x f_y\nn = [1, 0, f_x] × [0, 1, f_y]\n```\n\n::: {.cell-output .cell-output-display execution_count=6}\n```\n3-element Vector{Sym}:\n -fₓ\n -f_y\n 1\n```\n:::\n:::\n\n\nLet $\\vec{x} = \\langle a, b, f(a,b)$. The tangent plane at $\\vec{x}$ then is described by all vectors $\\vec{v}$ with $\\vec{n}\\cdot(\\vec{v} - \\vec{x}) = 0$. Using $\\vec{v} = \\langle x,y,z\\rangle$, we have:\n\n\n\n$$\n[-\\frac{\\partial f}{\\partial x}, -\\frac{\\partial f}{\\partial y}, 1] \\cdot [x-a, y-b, z - f(a,b)] = 0,\n$$\n\n\nor,\n\n\n\n$$\nz = f(a,b) + \\frac{\\partial f}{\\partial x} (x-a) + \\frac{\\partial f}{\\partial y} (y-b),\n$$\n\n\nwhich is more compactly expressed as\n\n\n\n$$\nz = f(a,b) + \\nabla(f) \\cdot \\langle x-a, y-b \\rangle.\n$$\n\n\nThis form would then generalize to scalar functions from $R^n \\rightarrow R$. This is consistent with the definition of $f$ being differentiable, where $\\nabla{f}$ plays the role of the slope in the formulas.\n\n\nThe following figure illustrates the above for the function $f(x,y) = 6 - x^2 - y^2$:\n\n::: {.cell hold='true' execution_count=6}\n``` {.julia .cell-code}\nf(x,y) = 6 - x^2 -y^2\nf(x)= f(x...)\n\na,b = 1, -1/2\n\n\n# draw surface\nxr = 7/4\nxs = ys = range(-xr, xr, length=100)\nsurface(xs, ys, f, legend=false)\n\n# visualize tangent plane as 3d polygon\npt = [a,b]\ntplane(x) = f(pt) + gradient(f)(pt) ⋅ (x - [a,b])\n\npts = [[a-1,b-1], [a+1, b-1], [a+1, b+1], [a-1, b+1], [a-1, b-1]]\nplot!(unzip([[pt..., tplane(pt)] for pt in pts])...)\n\n# plot paths in x and y direction through (a,b)\nγ_x(t) = pt + t*[1,0]\nγ_y(t) = pt + t*[0,1]\n\nplot_parametric!((-xr-a)..(xr-a), t -> [γ_x(t)..., (f∘γ_x)(t)], linewidth=3)\nplot_parametric!((-xr-b)..(xr-b), t -> [γ_y(t)..., (f∘γ_y)(t)], linewidth=3)\n\n# draw directional derivatives in 3d and normal\npt = [a, b, f(a,b)]\nfx, fy = gradient(f)(a,b)\narrow!(pt, [1, 0, fx], linewidth=3)\narrow!(pt, [0, 1, fy], linewidth=3)\narrow!(pt, [-fx, -fy, 1], linewidth=3) # normal\n\n# draw point in base, x-y, plane\npt = [a, b, 0]\nscatter!(unzip([pt])...)\narrow!(pt, [1,0,0], linestyle=:dash)\narrow!(pt, [0,1,0], linestyle=:dash)\n```\n\n::: {.cell-output .cell-output-display execution_count=7}\n![](scalar_functions_applications_filLine truncated
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