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pluto_calculus_II.jl
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### A Pluto.jl notebook ###
# v0.20.3
using Markdown
using InteractiveUtils
# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
#! format: off
quote
local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
local el = $(esc(element))
global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
el
end
#! format: on
end
# ╔═╡ 5ffc5ec0-f91f-11ec-0552-37ef5f25102d
begin
using PlutoUI
using Plots
using Symbolics
using ForwardDiff
using LaTeXStrings
using Images
using TestImages
using ImageView
using LinearAlgebra
# using FFTW
# using DSP
# using Gaston
nbsp = html" "
end
# ╔═╡ edd96a98-2825-4149-aed5-a9d00767a65f
md"""
### Planes in $\mathbb{R}^3$
"""
# ╔═╡ 7abdbf5c-5ed2-41c8-8e37-64d80a3b4203
md"""
α = $(@bind a0 Slider(-5:0.5:5, show_value=true, default=1))
$nbsp $nbsp
β = $(@bind b0 Slider(-5:0.5:5, show_value=true, default=1))
altitude = $(@bind latitude0 Slider(0:2:88, show_value=true, default=30)) deg
$nbsp $nbsp
azimuth= $(@bind longitude0 Slider(0:2:358, show_value=true, default=30)) deg
"""
# ╔═╡ 3bb5a21a-90fe-44a5-b3e9-9c380e1d2dd9
begin
x1 = range(-1, length=21, stop=1);
y1 = range(-1, length=21, stop=1);
h0=zeros(length(x1),length(y1));
for k=1:21
for l=1:21
h0[k,l]=a0*x1[k]+b0*y1[l];
end
end
end
# ╔═╡ c3d123e4-e88f-46f6-99a8-4f57a934bd8b
begin
wireframe(x1,y1,h0,
size=(800,600),
grid=false,
zlim=(-5,5),
title=L"z=αx+βy",
xticks=([-1 0 1;],[-1,0,1]),
yticks=([-1 0 1;],[-1,0,1]),
zticks=([-5 -4 -3 -2 -1 0 1 2 3 4 5;],[-5,-4,-3,-2,-1,0,1,2,3,4,5]),
xlabel="\$x\$",
ylabel="\$y\$",
#color=cgrad(:bwr),
camera = (longitude0, latitude0))
scatter!([0],[0],[0],color=:magenta,label="Origin",markersize=4)
end
# ╔═╡ 22ef50ed-99c8-4363-8de9-3253995b71cf
md"""
### Tangent Plane of a Surface
"""
# ╔═╡ ebc4b499-b919-418f-a2ad-52b1b061982a
md"""
a = $(@bind a1 Slider(-0.7:0.1:0.7, show_value=true, default=-0.3))
$nbsp $nbsp
b = $(@bind b1 Slider(-0.7:0.1:0.7, show_value=true, default=-0.3))
altitude = $(@bind latitude Slider(0:2:88, show_value=true, default=30)) deg
$nbsp $nbsp
azimuth= $(@bind longitude Slider(0:2:358, show_value=true, default=30)) deg
"""
# ╔═╡ bdcefd34-2c55-4eea-aea3-54df8739d6ab
begin
p1=a1;
q1=b1;
#k0=Int64(float(50*(a1+0.7))+15);
#l0=Int64(float(50*(b1+0.7))+15);
r1=1-a1^2-b1^2;
f1=zeros(length(x1),length(y1));
h1=zeros(length(x1),length(y1));
for k=1:21
for l=1:21
f1[k,l]=1-x1[k]^2-y1[l]^2;
h1[k,l]=1+a1^2+b1^2-2*b1*x1[k]-2*a1*y1[l];
end
end
end
# ╔═╡ f6e03f57-4132-4e08-88b4-4c30efb68c89
begin
wireframe(x1,y1,f1,
size=(800,600),
#grid=false,
zlim=(-1,3),
title="\$z=f(x,y)\$ and \$z=h(x,y)\$",
xticks=([-1 0 1;],[-1,0,1]),
yticks=([-1 0 1;],[-1,0,1]),
zticks=([-1 0 1 2 3;],[-1,0,1,2,3]),
xlabel="\$x\$",
ylabel="\$y\$",
#color=cgrad(:bwr),
camera = (longitude, latitude))
surface!(x1,y1,h1,
alpha=0.7,
colorbar=false)
scatter!([p1],[q1],[r1],color=:magenta,label="tangent point",markersize=4)
end
# ╔═╡ 147e9b2d-9149-40f9-a091-d4db86c85d58
# ╔═╡ b52a7ba4-d26c-4973-ac65-d3c062381a0c
md"""
### Vector Fields on $\mathbb{R}^2$
$\vec{v}(x,y)
=
\begin{bmatrix}
v_1(x,y) \\ v_2(x,y)
\end{bmatrix}
:=
\begin{bmatrix}
a & b
\\
c & d
\end{bmatrix}
\begin{bmatrix}
x \\ y
\end{bmatrix}
=
\begin{bmatrix}
ax+by \\ cx+dy
\end{bmatrix}.$
"""
# ╔═╡ dcbaa68e-ab73-4973-a4a2-9f8e5f6641fd
md"""
a = $(@bind a11 Slider(-3:0.1:3, show_value=true, default=0))
$nbsp
b = $(@bind a12 Slider(-3:0.1:3, show_value=true, default=-1))
c = $(@bind a21 Slider(-3:0.1:3, show_value=true, default=1))
$nbsp
d = $(@bind a22 Slider(-3:0.1:3, show_value=true, default=0))
"""
# ╔═╡ 17357173-3844-4412-8882-d2faf02f90cc
begin
xs = -2:0.2:2
ys = -2:0.2:2
df(x, y) = normalize([a11*x+a12*y, a21*x+a22*y]) ./ 10
xxs = [x for x in xs for y in ys]
yys = [y for x in xs for y in ys]
quiver(xxs, yys, quiver=df,
title=L"(ax+by,cx+dy)",
xlim=(-2,2),
ylim=(-2,2),
xlabel=L"x",
ylabel=L"y",
aspect_ratio = :equal)
end
# ╔═╡ be1d8798-c5ec-48cb-96e7-c42bb31dba29
# ╔═╡ 0440bbe2-4865-4fd8-bb6d-5ec6d2eaa480
md"""
### 2D Polar Coordinate System
"""
# ╔═╡ d8691762-85d5-4cac-9fc3-e492c918e75e
md"""
$r$ = $(@bind r2 Slider(2:0.1:5, show_value=true, default=4))
$θ$ = $(@bind θ2 Slider(0:1:359, show_value=true, default=45)) degree
"""
# ╔═╡ ca5f6714-8593-480f-8a2e-57cdb71cfcb9
begin
theta2=range(0,359, length=360);
x2=r2*cos.(2*pi*theta2/360);
y2=r2*sin.(2*pi*theta2/360);
for i=1:1
aaa=1;
end
end
# ╔═╡ 70c8eaff-c896-4bff-be2f-effefb8a437b
begin
plot([0,r2*cos(2*pi*θ2/360)],[0,r2*sin(2*pi*θ2/360)],
size=(600,600),
grid=false,
xlim=(-5.3,5.3),
ylim=(-5.3,5.3),
label=false,
xaxis=false,
yaxis=false,
linewidth=3,
linecolor=:blue,
aspect_ratio=1.0)
scatter!((r2*cos(2*pi*θ2/360),r2*sin(2*pi*θ2/360)),
markersize=4,color=:magenta,label="(rcosθ,rsinθ)")
plot!(x2[1:θ2],y2[1:θ2],label="θ",linewidth=2)
plot!([-5,5],[0,0],label=false,
arrow=3,linewidth=1,linecolor=:black,
annotations=[(5.4, 0, "\$x\$", 16)])
plot!([0,0],[-5,5],label=false,
arrow=3,linewidth=1,linecolor=:black,
annotations=[(0,5.4, "\$y\$", 16)])
plot!([r2*cos(2*pi*θ2/360),r2*cos(2*pi*θ2/360)],[0,r2*sin(2*pi*θ2/360)],label=false,linewidth=1,linecolor=:black, linestyle =:dash)
plot!([0,r2*cos(2*pi*θ2/360)],[r2*sin(2*pi*θ2/360),r2*sin(2*pi*θ2/360)],label=false,linewidth=1,linecolor=:black, linestyle =:dash)
end
# ╔═╡ b84cacfd-7072-4aeb-8435-f8b30c3f43fb
md"""
### 3D Polar Coordinate System
"""
# ╔═╡ 4c217a39-1d98-4025-8eb1-6a1a7f1d0142
md"""
$r$ = $(@bind r3 Slider(2:0.1:5, show_value=true, default=4))
$θ$ = $(@bind θ3 Slider(0:1:179, show_value=true, default=30)) degree
$φ$ = $(@bind φ3 Slider(0:1:359, show_value=true, default=45)) degree
"""
# ╔═╡ 1ec0ef68-2b28-485d-9a09-765984ae0f7e
begin
theta3=range(0,179, length=180);
varphi3=range(0,359, length=360);
x3a=r3*sin.(pi*theta3/180)*cos(2*pi*φ3/360);
y3a=r3*sin.(pi*theta3/180)*sin(2*pi*φ3/360);
z3a=r3*cos.(pi*theta3/180);
x3b=r3*sin(pi*θ3/180)*cos.(2*pi*varphi3/360);
y3b=r3*sin(pi*θ3/180)*sin.(2*pi*varphi3/360);
z3b=r3*zeros(360);
for i=1:1
aaaa=1;
end
end
# ╔═╡ f044b5d4-8785-4cc2-9e0b-bd91fe962316
begin
plot3d([0,r3*sin(pi*θ3/180)*cos(2*pi*φ3/360)],
[0,r3*sin(pi*θ3/180)*sin(2*pi*φ3/360)],
[0,r3*cos(pi*θ3/180)],
size=(600,600),
grid=false,
xlim=(-5.5,5.5),
ylim=(-5.5,5.5),
zlim=(-5.5,5.5),
label=false,
xaxis=false,
yaxis=false,
zaxis=false,
linewidth=3,
linecolor=:blue,
aspect_ratio=1.0,
legend=:topleft,
camera=(60,30))
plot3d!([0,r3*sin(pi*θ3/180)*cos(2*pi*φ3/360)],
[0,r3*sin(pi*θ3/180)*sin(2*pi*φ3/360)],
[0,0],
label=false,
linewidth=3,
linecolor=:magenta)
scatter3d!((r3*sin(pi*θ3/180)*cos(2*pi*φ3/360),
r3*sin(pi*θ3/180)*sin(2*pi*φ3/360),
r3*cos(pi*θ3/180)),
markersize=4,color=:red,label="(rsinθcosφ,rsinθsinφ,rcosθ)")
scatter3d!((r3*sin(pi*θ3/180)*cos(2*pi*φ3/360),
r3*sin(pi*θ3/180)*sin(2*pi*φ3/360),
0),
markersize=4,color=:green,label="(rsinθcosφ,rsinθsinφ,0)")
plot3d!(x3a[1:θ3],y3a[1:θ3],z3a[1:θ3],label="θ",linewidth=2,linecolor=:skyblue)
plot3d!(x3b[1:φ3],y3b[1:φ3],z3b[1:φ3],label="φ",linewidth=2,linecolor=:lightgreen)
plot3d!([-5,5],[0,0],[0,0],label=false,
arrow=3,linewidth=1,linecolor=:black,
annotations=[(5.6, 0, 0, "\$x\$", 16)])
plot3d!([0,0],[-5,5],[0,0],label=false,
arrow=3,linewidth=1,linecolor=:black,
annotations=[(0, 5.6, 0, "\$y\$", 16)])
plot3d!([0,0],[0,0],[-5,5],label=false,
arrow=3,linewidth=1,linecolor=:black,
annotations=[(0, 0, 5.6, "\$z\$", 16)])
plot3d!([0,r3*sin(pi*θ3/180)*cos(2*pi*φ3/360)],
[0,r3*sin(pi*θ3/180)*sin(2*pi*φ3/360)],
[r3*cos(pi*θ3/180),r3*cos(pi*θ3/180)],
label=false,linewidth=1,linecolor=:black, linestyle =:dash)
plot3d!([r3*sin(pi*θ3/180)*cos(2*pi*φ3/360),r3*sin(pi*θ3/180)*cos(2*pi*φ3/360)],
[r3*sin(pi*θ3/180)*sin(2*pi*φ3/360),r3*sin(pi*θ3/180)*sin(2*pi*φ3/360)],
[0,r3*cos(pi*θ3/180)],
label=false,linewidth=1,linecolor=:black, linestyle =:dash)
plot3d!([r3*sin(pi*θ3/180)*cos(2*pi*φ3/360),r3*sin(pi*θ3/180)*cos(2*pi*φ3/360)],
[0,r3*sin(pi*θ3/180)*sin(2*pi*φ3/360)],
[0,0],
label=false,linewidth=1,linecolor=:black, linestyle =:dash)
plot3d!([0,r3*sin(pi*θ3/180)*cos(2*pi*φ3/360)],
[r3*sin(pi*θ3/180)*sin(2*pi*φ3/360),r3*sin(pi*θ3/180)*sin(2*pi*φ3/360)],
[0,0],
label=false,linewidth=1,linecolor=:black, linestyle =:dash)
end
# ╔═╡ 9d916a08-0b7a-48dc-a68f-5f947908d4bd
# ╔═╡ 2a3e1a6d-bf0a-4ca5-80bd-5f4eb928f23f
begin
N4=100;
x4 = range(-1.7, length=27, stop=0.9);
y4 = range(-1.3, length=27, stop=1.3);
f4=zeros(length(x4),length(y4));
for k=1:27
for l=1:27
f4[l,k]=2*x4[k]^3+x4[k]*y4[l]^2+5*x4[k]^2+y4[l]^2;
end
end
end
# ╔═╡ 4bd1098b-9048-46f6-af19-b6877775233b
md"""
### Newton's Method
Given a smooth funtion
$f:\mathbb{R}^n\ni\vec{x}=[x_1,\dotsc,x_n]^T \mapsto f(\vec{x})\in\mathbb{R}$
and an initial point $\vec{x}_0\in\mathbb{R}^n$,
Newton's method formally defines a sequence $\{\vec{x}_n\}_{n=1,2,3,\dotsc}$ by
$\vec{x}_{n+1}:=\vec{x}_n-\bigl(\nabla^2f(\vec{x}_n)\bigr)^{-1}{\nabla}f(\vec{x}_n),
\quad n=1,2,3,\dotsc.$
This is used for finding a critical point $\vec{x}$ such that $\nabla{f}(\vec{x})=\vec{0}$.
"""
# ╔═╡ 27cc7d0d-8e10-4592-bc86-a6807cb4afc3
md"""
$x_0$ = $(@bind a4 Slider(-1:0.1:1, show_value=true, default=0.8))
$y_0$ = $(@bind b4 Slider(-1:0.1:1, show_value=true, default=0.8))
$n$ = $(@bind n4 Slider(0:1:N4, show_value=true, default=0))
"""
# ╔═╡ 94fb423f-0da4-4c0f-ae3b-193de23d05e6
begin
X4=zeros(N4+1,3);
X4[1,1]=a4;
X4[1,2]=b4;
X4[1,3]=2*a4^3+a4*b4^2+5*a4^2+b4^2;
for n=1:N4
D=12*X4[n,1]^2-2*X4[n,2]^2+22*X4[n,1]+10;
X4[n+1,1]=(12*X4[n,1]^3-2*X4[n,1]*X4[n,2]^2+21*X4[n,1]^2+X4[n,2]^2+9*X4[n,1])/D;
X4[n+1,2]=X4[n,2]*(12*X4[n,1]^2-X4[n,2]^2+17*X4[n,1]+5)/D;
X4[n+1,3]=2*X4[n+1,1]^3+X4[n+1,1]*X4[n+1,2]^2+5*X4[n+1,1]^2+X4[n+1,2]^2;
end
end
# ╔═╡ b6e08e59-2e93-426c-89c1-da9869c3535e
begin
surface(x4,y4,f4,
size=(600,600),
grid=false,
xlim=(-1.7,0.9),
ylim=(-1.3,1.3),
zlim=(0,9),
alpha=0.8,
title="\$z=2x^3+xy^2+5x^2+y^2\$",
xticks=([-1 0 1;],[-1,0,1]),
yticks=([-1 0 1;],[-1,0,1]),
zticks=([0 5;],[0,5]),
xlabel="\$x\$",
ylabel="\$y\$",
colormap=:jet,
camera = (-20, 30),
legend = :none)
scatter!([0],[0],[0],color=:yellow,label=false,markersize=4)
scatter!([X4[1:n4+1,1]],[X4[1:n4+1,2]],[X4[1:n4+1,3]],color=:magenta,label=false,markersize=4)
end
# ╔═╡ bd8c13d8-d506-4f40-9298-dcd0d41497f0
begin
N5=100;
for i=1:1
end
end
# ╔═╡ ce9141fc-0a73-4f75-8cbc-ce4188001094
md"""
### Gradient Descent
Given a smooth funtion
$f:\mathbb{R}^n\ni\vec{x}=[x_1,\dotsc,x_n]^T \mapsto f(\vec{x})\in\mathbb{R}$
and an initial point $\vec{x}_0\in\mathbb{R}^n$,
the Gradient descent defines a sequence $\{\vec{x}_n\}_{n=1,2,3,\dotsc}$ by
$\vec{x}_{n+1}:=\vec{x}_n-s{\nabla}f(\vec{x}_n), \quad n=1,2,3,\dotsc,$
where $s=s_n>0$ is a small parameter called a step size or a learning rate, and can change at each $n$-th step. This is used for finding a critical point $\vec{x}$ such that $\nabla{f}(\vec{x})=\vec{0}$.
"""
# ╔═╡ eb62437b-bdcf-4353-b79e-30850f21a37b
md"""
$s$ = $(@bind s5 Slider(0.01:0.01:0.2, show_value=true, default=0.05))
$x_0$ = $(@bind a5 Slider(-1:0.1:1, show_value=true, default=0.8))
$y_0$ = $(@bind b5 Slider(-1:0.1:1, show_value=true, default=0.8))
$n$ = $(@bind n5 Slider(0:1:N5, show_value=true, default=0))
"""
# ╔═╡ 0029ed3e-594e-4a98-b790-212b43c62b07
begin
X5=zeros(N5+1,3);
X5[1,1]=a5;
X5[1,2]=b5;
X5[1,3]=2*a5^3+a5*b5^2+5*a5^2+b5^2;
for n=1:N5
X5[n+1,1]=X5[n,1]-s5*(6*X5[n,1]^2+X5[n,2]^2+10*X5[n,1]);
X5[n+1,2]=X5[n,2]-s5*(2*X5[n,1]*X5[n,2]+2*X5[n,2]);
X5[n+1,3]=2*X5[n+1,1]^3+X5[n+1,1]*X5[n+1,2]^2+5*X5[n+1,1]^2+X5[n+1,2]^2;
end
end
# ╔═╡ 269237a6-fe05-4839-bf8f-990b23ab5c02
begin
surface(x4,y4,f4,
size=(600,600),
grid=false,
xlim=(-1.7,0.9),
ylim=(-1.2,1.2),
zlim=(0,9),
alpha=0.7,
title="\$z=2x^3+xy^2+5x^2+y^2\$",
xticks=([-1 0 1;],[-1,0,1]),
yticks=([-1 0 1;],[-1,0,1]),
zticks=([0 5;],[0,5]),
xlabel="\$x\$",
ylabel="\$y\$",
colormap=:jet,
camera = (-20, 30),
legend = :none)
scatter!([0],[0],[0],color=:yellow,label=false,markersize=4)
scatter!([X5[1:n5+1,1]],[X5[1:n5+1,2]],[X5[1:n5+1,3]],color=:magenta,label=false,markersize=4)
end
# ╔═╡ 44ff650e-f21c-4e66-a1be-9a28c860eb54
md"""
### Newton's Method vs Gradient Descent
Given a smooth funtion
$f:\mathbb{R}^n\ni\vec{x}=[x_1,\dotsc,x_n]^T \mapsto f(\vec{x})\in\mathbb{R}$
and an initial point $\vec{x}_0\in\mathbb{R}^n$,
- Newton's method formally defines a sequence $\{\vec{x}_n\}_{n=1,2,3,\dotsc}$ by
$\vec{x}_{n+1}:=\vec{x}_n-\bigl(\nabla^2f(\vec{x}_n)\bigr)^{-1}{\nabla}f(\vec{x}_n),
\quad n=1,2,3,\dotsc.$
- The Gradient descent defines a sequence $\{\vec{x}_n\}_{n=1,2,3,\dotsc}$ by
$\vec{x}_{n+1}:=\vec{x}_n-s{\nabla}f(\vec{x}_n), \quad n=1,2,3,\dotsc,\quad s>0.$
"""
# ╔═╡ c628098d-9052-4729-aabb-55fb4d0de037
# ╔═╡ c496505d-f561-4f0f-8a25-a53a01e56c85
md"""
learning rate $s$ = $(@bind s6 Slider(0.01:0.01:0.2, show_value=true, default=0.05))
$x_0$ = $(@bind a6 Slider(-1:0.1:1, show_value=true, default=0.8))
$y_0$ = $(@bind b6 Slider(-1:0.1:1, show_value=true, default=0.8))
steps $n$ = $(@bind n6 Slider(0:1:N5, show_value=true, default=0))
"""
# ╔═╡ c16941ca-a975-443b-a277-a3b91b7a99b4
begin
Y4=zeros(N4+1,3);
Y4[1,1]=a6;
Y4[1,2]=b6;
Y4[1,3]=2*a6^3+a6*b6^2+5*a6^2+b6^2;
for n=1:N4
D6=12*Y4[n,1]^2-2*Y4[n,2]^2+22*Y4[n,1]+10;
Y4[n+1,1]=(12*Y4[n,1]^3-2*Y4[n,1]*Y4[n,2]^2+21*Y4[n,1]^2+Y4[n,2]^2+9*Y4[n,1])/D6;
Y4[n+1,2]=Y4[n,2]*(12*Y4[n,1]^2-Y4[n,2]^2+17*Y4[n,1]+5)/D6;
Y4[n+1,3]=2*Y4[n+1,1]^3+Y4[n+1,1]*Y4[n+1,2]^2+5*Y4[n+1,1]^2+Y4[n+1,2]^2;
end
Y5=zeros(N4+1,3);
Y5[1,1]=a6;
Y5[1,2]=b6;
Y5[1,3]=2*a6^3+a6*b6^2+5*a6^2+b6^2;
for n=1:N5
Y5[n+1,1]=Y5[n,1]-s6*(6*Y5[n,1]^2+Y5[n,2]^2+10*Y5[n,1]);
Y5[n+1,2]=Y5[n,2]-s6*(2*Y5[n,1]*Y5[n,2]+2*Y5[n,2]);
Y5[n+1,3]=2*Y5[n+1,1]^3+Y5[n+1,1]*Y5[n+1,2]^2+5*Y5[n+1,1]^2+Y5[n+1,2]^2;
end
end
# ╔═╡ e3e95253-3057-4711-8d96-4d1933a484c5
begin
surface(x4,y4,f4,
size=(600,600),
grid=false,
xlim=(-1.7,0.9),
ylim=(-1.3,1.3),
zlim=(0,9),
alpha=0.8,
title="Newton's Method vs Gradient Descent",
xticks=([-1 0 1;],[-1,0,1]),
yticks=([-1 0 1;],[-1,0,1]),
zticks=([0 5;],[0,5]),
xlabel="\$x\$",
ylabel="\$y\$",
colormap=:jet,
colorbar=false,
camera = (-20, 30))
scatter!([0],[0],[0],color=:yellow,label="Minimal Point",markersize=7)
scatter!([Y4[1:n6+1,1]],[Y4[1:n6+1,2]],[Y4[1:n6+1,3]],color=:magenta,label="Newton's Method",markersize=4)
scatter!([Y5[1:n6+1,1]],[Y5[1:n6+1,2]],[Y5[1:n6+1,3]],color=:green,label="Gradient Descent",markersize=4)
scatter!([Y4[1:1,1]],[Y4[1:1,2]],[Y4[1:1,3]],color=:red,label="Starting Point",markersize=6)
end
# ╔═╡ da6f2cd6-0009-46a9-a1da-5b44ce4eee90
md"""
### Riemann Sum
"""
# ╔═╡ 9cdaa699-ba6d-4780-8e7c-6af3b9ca0e6c
begin
n8=5;
N8=100;
x8=range(-0.3, length=N8+61, stop=1.3);
y8=range(-0.3,length=N8+61, stop=1.3);
R8=rand(Float64, (n8,n8));
z8=zeros(N8+61,N8+61);
z8[30,30]=R8[1,1];
for k=1:N8
z8[30,k+30]=R8[1,Int64(ceil(k/20))];
z8[k+30,30]=R8[Int64(ceil(k/20)),1];
for l=1:N8
z8[k+30,l+30]=R8[Int64(ceil(k/20)),Int64(ceil(l/20))];
end
end
surface(x8,y8,z8,
title="A Step Function",
size=(800,600),
colormap=:GnBu,
grid=false,
zlim=(0,1.1),
xticks=([0 1;],[L"a", L"b"]),
yticks=([0 1;],[L"c", L"d"]),
zticks=false,
xlabel=L"x",
ylabel=L"y",
zlabel=L"z",
legend = :none)
plot3d!([-0.3,1.3],[0,0],[0,0],
label=false,linewidth=1,linecolor=:black, linestyle =:dash)
plot3d!([1,1.3],[1,1],[0,0],
label=false,linewidth=1,linecolor=:black, linestyle =:dash)
plot3d!([1,1],[-0.3,1.3],[0,0],
label=false,linewidth=1,linecolor=:black, linestyle =:dash)
plot3d!([0,0],[-0.3,0],[0,0],
label=false,linewidth=1,linecolor=:black, linestyle =:dash)
end
# ╔═╡ 47259765-a80e-411a-b753-4d3c9e98dd08
begin
N9=100;
R9=zeros(N9,N9);
for j=1:N9
for k=1:N9
R9[j,k]=Float64(1-(1/2-j/N9)^2-(1/2-k/N9)^2)
end
end
end
# ╔═╡ 03d68283-a3e6-4525-bcdf-a850231b69db
md"""
### Convergence of Riemann Sum
Given a continuous function $f(x,y)$ on $I=[a,b]\times[c,d]$. Theoretically $f(x,y)$ is Riemann integrable.
We have a division of $I$ splitting $[a,b]$ and $[c,d]$ into $n$-equal parts respectively:
$I_{jk}=\left[a+\frac{(b-a)(j-1)}{n},a+\frac{(b-a)j}{n}\right]\times\left[c+\frac{(d-c)(k-1)}{n},c+\frac{(d-c)k}{n}\right].$
Pick up arbitrary $(x_{jk},y_{jk}) \in I_{jk}$. Define a Riemann sum $R_n$ of $f(x,y)$ by
$R_n
=
\sum_{j=1}^n\sum_{k=1}^nf(x_{jk},y_{jk})\lvert{I_{jk}}\rvert
=
\frac{(b-a)(d-c)}{n^2}
\sum_{j=1}^n\sum_{k=1}^nf(x_{jk},y_{jk}).$
Then
$R_n \rightarrow \iint_{I}f(x,y)dxdy \quad (n \rightarrow \infty).$
"""
# ╔═╡ 5c399ee3-c07f-4f81-9030-482e027365a8
md"""
$n$ = $(@bind n9 Slider(1:1:50, show_value=true, default=1))
"""
# ╔═╡ 27d388a0-4be6-4918-b8f4-fec391114b0b
begin
x9=range(-0.3, length=N9+61, stop=1.3);
y9=range(-0.3,length=N9+61, stop=1.3);
z9=zeros(N9+61,N9+61);
z9[30,30]=R9[1,1];
for k=1:N9
z9[30,k+30]=R9[1,Int64(ceil(ceil(n9*k/100)*N9/n9))];
z9[k+30,30]=R9[Int64(ceil(ceil(n9*k/100)*N9/n9)),1];
for l=1:N9
z9[k+30,l+30]=R9[Int64(ceil(ceil(n9*k/N9)*N9/n9)),Int64(ceil(ceil(n9*l/N9)*N9/n9))];
end
end
surface(x9,y9,z9,
title="Convergence of Riemann Sum",
size=(800,600),
colormap=:GnBu,
grid=false,
zlim=(0,1.1),
xticks=([0 1;],[L"a", L"b"]),
yticks=([0 1;],[L"c", L"d"]),
zticks=false,
xlabel=L"x",
ylabel=L"y",
zlabel=L"z",
legend = :none)
plot3d!([-0.3,1.3],[0,0],[0,0],
label=false,linewidth=1,linecolor=:black, linestyle =:dash)
plot3d!([1,1.3],[1,1],[0,0],
label=false,linewidth=1,linecolor=:black, linestyle =:dash)
plot3d!([1,1],[-0.3,1.3],[0,0],
label=false,linewidth=1,linecolor=:black, linestyle =:dash)
plot3d!([0,0],[-0.3,0],[0,0],
label=false,linewidth=1,linecolor=:black, linestyle =:dash)
end
# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210"
ImageView = "86fae568-95e7-573e-a6b2-d8a6b900c9ef"
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LaTeXStrings = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f"
LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
Symbolics = "0c5d862f-8b57-4792-8d23-62f2024744c7"
TestImages = "5e47fb64-e119-507b-a336-dd2b206d9990"
[compat]
ForwardDiff = "~0.10.35"
ImageView = "~0.11.5"
Images = "~0.25.3"
LaTeXStrings = "~1.3.0"
Plots = "~1.38.15"
PlutoUI = "~0.7.51"
Symbolics = "~5.5.0"
TestImages = "~1.7.1"
"""
# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
# This file is machine-generated - editing it directly is not advised
julia_version = "1.11.1"
manifest_format = "2.0"
project_hash = "d88f4c6502d78ad965bae9eb8f7191cde9c863c2"
[[deps.ADTypes]]
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[[deps.ATK_jll]]
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version = "2.38.0+0"
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git-tree-sha1 = "c3c29bf6363b3ac3e421dc8b2ba8e33bdacbd245"
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[[deps.AbstractPlutoDingetjes]]
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[deps.ArrayInterface.weakdeps]
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