### A Pluto.jl notebook ###
# v0.17.7

using Markdown
using InteractiveUtils

# ╔═╡ 6bcd5cd3-57c0-40a6-83eb-87ace39d50f9
begin
	using Plots
	using ForwardDiff
	using LaTeXStrings

	md"""
	# Homework 3

	Eric Nguyen
	"""
end

# ╔═╡ 9ce3d4d6-4a66-46ab-895e-13e03cd3990e
md"""
### Problem 1

The Taylor expansion of a function ``f`` about ``x_0`` is defined as

$f(x) = \sum_{n = 0}^∞ \frac{1}{n!} f^{(n)}(x_0)(x - x_0)^n$

First, we want to find the first three nonzero terms of the Taylor expansions about ``x = 0`` for ``\cosh{(x)}``.

$\cosh(x) = 1 + \frac{1}{2!} x^2 + \frac{1}{4!} x^4 + ⋯$

Next, for ``\sinh{(x)}``.

$\sinh(x) = x + \frac{1}{3!} x^3 + \frac{1}{5!} x^5 + ⋯$
"""

# ╔═╡ 05029d3c-bde7-4548-9f1d-9fd981e217d8
let
	f(x) = cosh(x)
	plot(f, label=L"\cosh(x)", title="Taylor expansion about $(L"x = 0") for $(L"\cosh{(x)}")")
	plot!(x -> 1, label="First term")
	plot!(x -> 1 + x^2 / factorial(2), label="Second term")
	plot!(x -> 1 + x^2 / factorial(2) + x^4 / factorial(4), label="Third term")
end

# ╔═╡ 9f7db206-54f8-4374-85af-00dc38ec00d6
let
	f(x) = sinh(x)
	plot(f, label=L"\sinh(x)", title="Taylor expansion about $(L"x = 0") for $(L"\sinh{(x)}")")
	plot!(x -> x, label="First term")
	plot!(x -> x + x^3 / factorial(3), label="Second term")
	plot!(x -> x + x^3 / factorial(3) + x^5 / factorial(5), label="Third term")
end

# ╔═╡ 751fa195-5e94-4f19-9b91-56c599d8c448
md"""
### Problem 2

**(a).**
For the point charge at ``+a/2``, the electric field at some ``z`` is

$E(z) = \frac{kq}{\left(z - \frac{a}{2}\right)^2}$

For the point charge at ``-a/2``, the electric field at some ``z`` is

$E(z) = -\frac{kq}{\left(z + \frac{a}{2}\right)^2}$

Then the electric field at some ``z`` by both point charges is

$\begin{align*}
E(z) &= \frac{kq}{\left(z - \frac{a}{2}\right)^2} - \frac{kq}{\left(z + \frac{a}{2}\right)^2} \\
&= \frac{kq \left[\left(z + \frac{a}{2}\right)^2 - \left(z - \frac{a}{2}\right)^2\right]}{\left(z - \frac{a}{2}\right)^2\left(z + \frac{a}{2}\right)^2} \\
&= \frac{kq\left(z^2 + za + \frac{a^2}{4} - z^2 + za - \frac{a^2}{4}\right)}{\left(z - \frac{a}{2}\right)^2\left(z + \frac{a}{2}\right)^2} \\
&= \frac{kq\left(2za\right)}{\left(z - \frac{a}{2}\right)^2\left(z + \frac{a}{2}\right)^2} \\
&≈ \frac{2zpk}{z^4} = \frac{1}{4\pi\epsilon_0} \frac{2p}{z^3}
\end{align*}$

The electric field from a dipole falls off faster since it decreases proportionally with ``z^3`` as opposed to it decreasing proportionally with ``z`` for an isolated electric charge.

**(b).**
The distance from the point charges to some ``x`` is given by

$r = \sqrt{x^2 + \frac{a^2}{4}}$

Then the electric field for a position on the ``x``-axis in the direction of the negative point charge is given by the parallel component of each electric field,

$E(x) = \frac{2kq}{x^2 + \frac{a^2}{4}} \cos{\theta}$

Using geometry, we can rewrite ``\cos{\theta}`` as

$\cos{\theta} = \frac{\frac{a}{2}}{r} = \frac{a}{2r}$

So ``E(x)`` is

$E(x) = \frac{kqa}{r \left(x^2 + \frac{a^2}{4}\right)} ≈ \frac{1}{4\pi\epsilon_0} \frac{p}{x^3}$

in the direction of the negative point charge.
The field from an electric dipole at some ``x`` position also falls off faster than that of an isolated electric charge.
"""

# ╔═╡ 7d31aca7-324c-4e17-8f0f-fc0dbbb92719
md"""
### Problem 3
"""

# ╔═╡ 2975571f-2562-4d63-a99e-fbb72a2c79d5
md"""
The value of ``x`` that maximizes ``f(x)`` is given by ``f'(x) = 0``,

$\frac{d}{dx} x^n e^{-x} = -x^n e^{-x} + nx^{n-1} e^{-x}$

$\begin{align*}
-x^n e^{-x} + nx^{n-1} e^{-x} &= 0 \\
-x^n + nx^{n-1} &= 0 \\
nx^{n-1} &= x^n \\
x &= n
\end{align*}$

The value of ``x`` that maximizes ``f(x)`` is ``n``.
"""

# ╔═╡ bc4fc99e-c732-4d21-8e75-226d1b0dac45
let
	n = 100
	f(x) = x^n * exp(-x)
	plot(f, xlims=[n-n/2,n+n/2])
end

# ╔═╡ 4e92fdc5-6f74-4f51-9d93-4821d86d00c8
md"""
$f(x) = (e^{\log{x}})^n e^{-e^{\log{x}}} = e^{n\log{x} - x}$

$y ≡ x-n \implies x = y+n$

$f(y) = e^{n\log{(y+n)} - n - y} = e^{n\log{n} + n \log{\left(1 + \frac{y}{n}\right)} - n - y}$

The Taylor series of the logarithm expression ``\log{\left(1 + \frac{y}{n}\right)}`` to second order about ``y = 0`` is

$\log{\left(1 + \frac{y}{n}\right)} = \log{(1)} + \frac{1}{n} (y) - \frac{1}{2} \frac{1}{n^2} (y)^2 = \frac{y}{n} - \frac{y^2}{2n^2}$

Gaussian integral

$∫_0^∞ x^n e^{-x} \,dx = n^n e^{-n} \sqrt{2 \pi n} = Γ(n + 1) ≈ n!$

In other words,

$n! ≈ \sqrt{2\pi n} \left(\frac{n}{e}\right)^n$
"""

# ╔═╡ 1dcd3bdc-7653-4ff7-ab41-03ae49ceb930
md"""
### Problem 4

To find ``\cos{3x}`` and ``\sin{3x}`` in terms of ``\cos{x}`` and ``\sin{x}`` we can apply Euler's formula and equate the real parts and imaginary parts.

$\begin{align*}
\cos{3x} + i \sin{3x} &= \cos{(x + x + x)} + i \sin{(x + x + x)} \\
&= (\cos{x} + i \sin{x})(\cos{x} + i \sin{x})(\cos{x} + i \sin{x}) \\
&= (\cos{x} + i \sin{x})^3 \qquad \text{Apply Binomial theorem} \\
&= (\cos{x})^3 + 3 (\cos{x})^2 (i \sin{x}) + 3 (\cos{x}) (i \sin{x})^2 + (i \sin{x})^3 \\
&= (\cos{x})^3 + 3 (\cos{x})^2 (i \sin{x}) - 3 (\cos{x}) (\sin{x})^2 - i(\sin{x})^3 \\
&= (\cos{x})^3 - 3 (\cos{x}) (\sin{x})^2 + 3 (\cos{x})^2 (i \sin{x}) - i(\sin{x})^3 \\
&= (\cos{x})^3 - 3 (\cos{x}) (\sin{x})^2 + i \left[3 (\cos{x})^2 (\sin{x}) - (\sin{x})^3\right] \\
\end{align*}$

Matching the real and imaginary parts, we see that

$\cos{3x} = (\cos{x})^3 - 3(\cos{x})(\sin{x})^2$

$\sin{3x} = 3(\cos{x})^2 (\sin{x}) - (\sin{x})^3$
"""

# ╔═╡ b61f60c0-967a-4b33-8590-87b77aa1f911
md"""
### Problem 5

For the first expression,

$e^{i (\pi/2)} = \cos{\frac{\pi}{2}} + i \sin{\frac{\pi}{2}} = i \implies i^i = (e^{i(\pi/2)})^i = e^{\pi/2}.$

For the second expression,

$\begin{align*}
\left(\frac{1 + i \sqrt{3}}{\sqrt{2} + i \sqrt{2}}\right)^{50} &= \left(\frac{2 \left(\frac{1}{2} + i \frac{\sqrt{3}}{2}\right)}{2 \left(\frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2}\right)}\right)^{50} \\
&= \left(\frac{\cos{\frac{\pi}{3}} + i \sin{\frac{\pi}{3}}}{\cos{\frac{\pi}{4}} + i \sin{\frac{\pi}{4}}}\right)^{50} \\
&= \left(\frac{e^{i\pi/3}}{e^{i\pi/4}}\right)^{50} \\
&= e^{50(i\pi/3 - i\pi/4)} \\
&= e^{50i(\pi/12)} \\
&= e^{i (25\pi/6)} \\
&= \cos{\frac{25\pi}{6}} + i \sin{\frac{25\pi}{6}}
\end{align*}$

For the third expression,

$\begin{align*}
\sinh{(1 + i\pi / 2)} &= \frac{e^{1 + i\pi/2} - e^{-(1 + i\pi/2)}}{2} \\
&= \frac{e^{2(i\pi/2+1)} - 1}{2e^{i\pi/2+1}} \\
&= \frac{e^{i\pi+2} - 1}{2e^{i\pi/2+1}} \\
\end{align*}$

For the fourth expression,

$\begin{align*}
y &= \tanh^{-1}{x} \\
\tanh{y} &= x \\
\frac{e^y - e^{-y}}{e^y + e^{-y}} &= x \\
e^y - e^{-y} &= xe^y + xe^{-y} \\
\frac{e^{2y} - 1}{e^y} &= \frac{xe^{2y} + x}{e^y} \\
e^{2y} - 1 &= xe^{2y} + x \\
e^{2y} - xe^{2y} &= 1 + x \\
e^{2y} (1 - x) &= 1 + x \\
e^{2y} &= \frac{1 + x}{1 - x} \\
2y &= \log{\left(\frac{1 + x}{1 - x}\right)} \\
y &= \frac{1}{2} \log{\left(\frac{1 + x}{1 - x}\right)}
\end{align*}$

$\begin{align*}
e^{2 \tanh^{-1}{i}} &= e^{2 ⋅ \frac{1}{2} \log{\left(\frac{1 + i}{1 - i}\right)}} \\
&= \frac{1 + i}{1 - i} \\
&= \frac{1 + e^{i\pi/2}}{1 - e^{i\pi/2}}
\end{align*}$
"""

# ╔═╡ 049ebeb2-cf5e-47b6-bd42-4f5371b3ec73
(1 + im) / (1 - im)

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[[deps.URIs]]
git-tree-sha1 = "97bbe755a53fe859669cd907f2d96aee8d2c1355"
uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4"
version = "1.3.0"

[[deps.UUIDs]]
deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"

[[deps.Unicode]]
uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"

[[deps.UnicodeFun]]
deps = ["REPL"]
git-tree-sha1 = "53915e50200959667e78a92a418594b428dffddf"
uuid = "1cfade01-22cf-5700-b092-accc4b62d6e1"
version = "0.4.1"

[[deps.Unzip]]
git-tree-sha1 = "34db80951901073501137bdbc3d5a8e7bbd06670"
uuid = "41fe7b60-77ed-43a1-b4f0-825fd5a5650d"
version = "0.1.2"

[[deps.Wayland_jll]]
deps = ["Artifacts", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "3e61f0b86f90dacb0bc0e73a0c5a83f6a8636e23"
uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89"
version = "1.19.0+0"

[[deps.Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "66d72dc6fcc86352f01676e8f0f698562e60510f"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.23.0+0"

[[deps.XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "1acf5bdf07aa0907e0a37d3718bb88d4b687b74a"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.9.12+0"

[[deps.XSLT_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "Pkg", "XML2_jll", "Zlib_jll"]
git-tree-sha1 = "91844873c4085240b95e795f692c4cec4d805f8a"
uuid = "aed1982a-8fda-507f-9586-7b0439959a61"
version = "1.1.34+0"

[[deps.Xorg_libX11_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll", "Xorg_xtrans_jll"]
git-tree-sha1 = "5be649d550f3f4b95308bf0183b82e2582876527"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.6.9+4"

[[deps.Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4e490d5c960c314f33885790ed410ff3a94ce67e"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.9+4"

[[deps.Xorg_libXcursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "12e0eb3bc634fa2080c1c37fccf56f7c22989afd"
uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724"
version = "1.2.0+4"

[[deps.Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fe47bd2247248125c428978740e18a681372dd4"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.3+4"

[[deps.Xorg_libXext_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "b7c0aa8c376b31e4852b360222848637f481f8c3"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
version = "1.3.4+4"

[[deps.Xorg_libXfixes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "0e0dc7431e7a0587559f9294aeec269471c991a4"
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8"
version = "5.0.3+4"

[[deps.Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.7.10+4"

[[deps.Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.4+4"

[[deps.Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.2+4"

[[deps.Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.10+4"

[[deps.Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.0+3"

[[deps.Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.13.0+3"

[[deps.Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.0+4"

[[deps.Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[deps.Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[deps.Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.2+4"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.27.0+4"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "cc4bf3fdde8b7e3e9fa0351bdeedba1cf3b7f6e6"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.0+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl", "OpenBLAS_jll"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "0.9.1+5"
"""

# ╔═╡ Cell order:
# ╟─6bcd5cd3-57c0-40a6-83eb-87ace39d50f9
# ╟─9ce3d4d6-4a66-46ab-895e-13e03cd3990e
# ╠═05029d3c-bde7-4548-9f1d-9fd981e217d8
# ╠═9f7db206-54f8-4374-85af-00dc38ec00d6
# ╟─751fa195-5e94-4f19-9b91-56c599d8c448
# ╟─7d31aca7-324c-4e17-8f0f-fc0dbbb92719
# ╟─2975571f-2562-4d63-a99e-fbb72a2c79d5
# ╠═bc4fc99e-c732-4d21-8e75-226d1b0dac45
# ╟─4e92fdc5-6f74-4f51-9d93-4821d86d00c8
# ╟─1dcd3bdc-7653-4ff7-ab41-03ae49ceb930
# ╟─b61f60c0-967a-4b33-8590-87b77aa1f911
# ╠═049ebeb2-cf5e-47b6-bd42-4f5371b3ec73
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002
