pluto_calculus_II.jl

### A Pluto.jl notebook ###
# v0.19.36

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)
    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
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
	
	nbsp = html"&nbsp"
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=6)) deg
 $nbsp $nbsp
 azimuth= $(@bind longitude Slider(0:2:358, show_value=true, default=30)) deg
"""

# ╔═╡ bdcefd34-2c55-4eea-aea3-54df8739d6ab
begin
	p1=b1;
	q1=a1;
	k0=Int64(float(50*(a1+0.7))+15);
	l0=Int64(float(50*(b1+0.7))+15);
	r1=1-a1^2-b1^2;
	x1 = range(-1, length=101, stop=1);
	y1 = range(-1, length=101, stop=1);
	f1=zeros(length(x1),length(y1));
	h1=zeros(length(x1),length(y1));
	for k=1:101
		for l=1:101
			f1[k,l]=1-x1[k]^2-y1[l]^2;
			h1[k,l]=1+a1^2+b1^2-2*a1*x1[k]-2*b1*y1[l];
		end
	end
end

# ╔═╡ f6e03f57-4132-4e08-88b4-4c30efb68c89
begin
    surface(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\$",
	        colormap=:jet,
		    camera = (longitude, latitude))
	surface!(x1,y1,h1,
		     colormap=:dense,
	         colorbar=false)
	scatter!([p1],[q1],[r1],color=:magenta,label="tangent point",markersize=4)
end

# ╔═╡ 10609a19-288b-4bc0-a349-0ae5589e9043
begin
    SEC1=plot(f1[:,l0],
		grid=false,
		linewidth=2,
		label="\$z=f(x,b)\$",
	    xticks=([1 51 101;],[-1,0,1]),
	    yticks=([0 1;],[0,1]),
	    xlabel="x",
	    ylabel="z")
	plot!(h1[:,l0],
		linewidth=2,
		label="\$z=h(x,b)\$")
	scatter!([k0],[h1[k0,l0]],
		     color=:magenta,
		     label="tangent point",
		    markersize=4)

    SEC2=plot(f1[k0,:],
		grid=false,
		linewidth=2,
		label="\$z=f(a,y)\$",
	    xticks=([1 51 101;],[-1,0,1]),
	    yticks=([0 1;],[0,1]),
	    xlabel="y",
	    ylabel="z")
	plot!(h1[k0,:],
		linewidth=2,
		label="\$z=h(a,y)\$")
	scatter!([l0],[h1[k0,l0]],
		     color=:magenta,
		     label="tangent point",
		    markersize=4)
	
    plot(SEC1,SEC2,layout=(1,2),size=(1200,400))
end

# ╔═╡ 147e9b2d-9149-40f9-a091-d4db86c85d58


# ╔═╡ 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

# ╔═╡ 851efeec-5a98-4e45-8c75-768920354537
md"""
### Newton mathod

$\vec{x}_{n+1}:=\vec{x}_n-\bigl(f^{\prime\prime}(\vec{x}_n)\bigr)^{-1}f^\prime(\vec{x}_n),\quad n=0,1,2,3,\dotsc$
"""

# ╔═╡ 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),
	        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, 50),
	        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(N4+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=(-1.3,1.3),
		    zlim=(0,9),
	        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, 50),
	        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"""
 $s$ = $(@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))

 $n$ = $(@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
    p6=surface(x4,y4,f4,
		    size=(800,600),
	        grid=false,
		    xlim=(-1.7,0.9),
		    ylim=(-1.3,1.3),
		    zlim=(0,9),
	        title="Newton's Method",
		    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, 50),
	        legend = :none)
	scatter!([0],[0],[0],color=:yellow,label=false,markersize=4)
	scatter!([Y4[1:n6+1,1]],[Y4[1:n6+1,2]],[Y4[1:n6+1,3]],color=:magenta,label=false,markersize=4)

	q6=surface(x4,y4,f4,
		    size=(800,600),
	        grid=false,
		    xlim=(-1.7,0.9),
		    ylim=(-1.3,1.3),
		    zlim=(0,9),
	        title="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,
		    camera = (-20, 50),
	        legend = :none)
	scatter!([0],[0],[0],color=:yellow,label=false,markersize=4)
	scatter!([Y5[1:n6+1,1]],[Y5[1:n6+1,2]],[Y5[1:n6+1,3]],color=:magenta,label=false,markersize=4)

	plot(p6,q6,layout=(1,2),size=(2000,800))
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 = """
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LaTeXStrings = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f"
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"""

# ╔═╡ 00000000-0000-0000-0000-000000000002
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deps = ["Artifacts", "Glib_jll", "JLLWrappers", "JpegTurbo_jll", "Libdl", "Libtiff_jll", "Pkg", "Xorg_libX11_jll", "libpng_jll"]
git-tree-sha1 = "e9190f9fb03f9c3b15b9fb0c380b0d57a3c8ea39"
uuid = "da03df04-f53b-5353-a52f-6a8b0620ced0"
version = "2.42.8+0"

[[deps.hicolor_icon_theme_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "b458a6f6fc2b1a8ca74ed63852e4eaf43fb9f5ea"
uuid = "059c91fe-1bad-52ad-bddd-f7b78713c282"
version = "0.17.0+3"

[[deps.iso_codes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "51559b9071db7e363047a34f658d495843ccd35c"
uuid = "bf975903-5238-5d20-8243-bc370bc1e7e5"
version = "4.11.0+0"

[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "3a2ea60308f0996d26f1e5354e10c24e9ef905d4"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.4.0+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"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.8.0+0"

[[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.libsixel_jll]]
deps = ["Artifacts", "JLLWrappers", "JpegTurbo_jll", "Libdl", "Pkg", "libpng_jll"]
git-tree-sha1 = "d4f63314c8aa1e48cd22aa0c17ed76cd1ae48c3c"
uuid = "075b6546-f08a-558a-be8f-8157d0f608a5"
version = "1.10.3+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"
version = "1.52.0+1"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+0"

[[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 = "9ebfc140cc56e8c2156a15ceac2f0302e327ac0a"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.4.1+0"
"""

# ╔═╡ Cell order:
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