Calculus and Kinematics

This document is try to use Julia to demonstrate the concepts related to the calculus and kinematics

• Loading the neccesary packages----------------

using Plots    #The package related to ploting the data
gr()           #The GR as plot background

Plots.GRBackend()

using SymPy   #The symbolic package

The limitation

Try to demonstrate the limit process

xi=0.01:0.1:1.01

0.01:0.1:1.01

yi=sin.(xi)

11-element Array{Float64,1}:
 0.009999833334166664
 0.10977830083717481 
 0.20845989984609956 
 0.3050586364434435  
 0.3986093279844229  
 0.48817724688290753 
 0.5728674601004813  
 0.6518337710215366  
 0.7242871743701426  
 0.7895037396899505  
 0.8468318446180152  

yi./xi

11-element Array{Float64,1}:
 0.9999833334166665
 0.9979845530652256
 0.9926661897433312
 0.9840601175594952
 0.9722178731327388
 0.9572102880057011
 0.9391269837712808
 0.9180757338331502
 0.8941816967532623
 0.8675865271318137
 0.838447370908926 

plot(xi,yi./xi,color=:black,linewidth=2,xlabel="x",ylabel="\$\\frac{\\sin(x)}{x}\$",legend=false)

x = symbols("x");
limit(sin(x)/x,x,0)

\begin{equation*}1\end{equation*}

xi=0.01:0.01:0.1;plot(xi,sin.(xi)./xi,color=:black,linewidth=2,xlabel="x",ylabel="\$\\frac{\\sin(x)}{x}\$",legend=false)

Differentiation

• Using discrete data to approach to the continuous limit(taking sin(x) as an example)----------------

plot(sin,0,3,linewidth=2,color=:cyan,lab="sin(x)")
x=0:1:3;plot!(x,sin.(x),linewidth=2,color=:black,lab="\\Delta x=1");
x=0:0.6:3;plot!(x,sin.(x),linewidth=2,color=:red,lab="\\Delta x=0.6");
x=0:0.5:3;plot!(x,sin.(x),linewidth=2,color=:blue,lab="\\Delta x=0.5");
x=0:0.2:3;plot!(x,sin.(x),linewidth=2,color=:yellow,lab="\\Delta x=0.2");
x=0:0.1:3;plot!(x,sin.(x),linewidth=2,color=:pink,lab="\\Delta x=0.1");
xlabel!("x");ylabel!("sin(x)")

• Demonstration of first dirivative as the slope----------------

plot(sin,0,π,lab="sin(x)",linewidth=2,color=:red)
Δ=0.5;
θ₀=π/3;
ds=(sin(θ₀+Δ)-sin(θ₀))/Δ;
plot!([0,π],ds.*([0,π].-θ₀).+sin(θ₀),linewidth=2,color=:blue,lab="Tagent line")

θ₀=π/3;anim=@animate for Δ in [1 0.8 0.6 0.4 0.2 0.1 0.001]
    plot(sin,0,π,lab="sin(x)",linewidth=2,color=:red,xlim=(0,π),ylim=(0,1.7))
    ds=(sin(θ₀+Δ)-sin(θ₀))/Δ;
    plot!([0,π],ds.*([0,π].-θ₀).+sin(θ₀),linewidth=2,color=:blue,lab="Tagent line")
    end;

gif(anim,"diff.gif",fps=2,show_msg=false)

θ = symbols("θ");diff(sin(θ))

\begin{equation*}\cos{\left(θ \right)}\end{equation*}

θ=π/3;Δθ=0.01:0.1:1.01

0.01:0.1:1.01

lim=sin.(θ.+Δθ).-sin.(θ)

11-element Array{Float64,1}:
 0.004956615757736871
 0.04965497771110228 
 0.08520416397625552 
 0.11124897883503404 
 0.12752919110663985 
 0.13388213429157803 
 0.13024433188154816 
 0.11665213159573395 
 0.09324134220640712 
 0.06024587658256764 
 0.01799541450987807 

plot(Δθ,lim./Δθ,w=2,shape=:circle)

Integration

• Demonstration of the integration

x=(0:0.01:1)π;plot(x,sin.(x),st=:bar,linewidth=2,color=:yellow,lab="\\Delta x=0.01\\pi");
plot!(sin,0,π,linewidth=2,color=:black)

• To demonstrate the numerical integration

xi=0.5;xe=2.5;Δx=0.00001;
x=xi:Δx:xe;
sx=sin.(exp.(x)); 
sum(sx.*Δx)

0.07894535615905263

integrate(sin(symbols("x")),xi,xe)

\begin{equation*}1.67872617743731\end{equation*}

The projectile motion

• Define the trajectory function

rx(t)=v.*cos(θ)*t

rx (generic function with 1 method)

ry(t)=v.*sin(θ)*t-g.*t.^2/2

ry (generic function with 1 method)

v=3;θ=π/3;g=9.8;

t=0:0.01:1;plot(rx(t),ry(t),lab="projectile",linewidth=2,color=:red)

p=plot(rx,ry,zeros(1),lab="projectile",linewidth=2,color=:blue,xlim=(-0.1,1.5),ylim=(-2,0.5))
traj=@animate for t=0:0.01:1
    push!(p,rx(t),ry(t))
    end;

gif(traj,"projectile.gif",fps=8,show_msg=false)

t=0:0.01:1;plot(rx.(t),ry.(t),lab="projectile",linewidth=2,color=:red,xlim=(-0.1,1.5),ylim=(-2.5,0.5))
traj_v=@animate for t=0.1:0.1:1
   plot!([0,rx(t)],[0,ry(t)],arrow=:true,linewidth=2,color=:blue,xlim=(-0.1,1.5),ylim=(-2.5,0.5))
    end;

gif(traj_v,"vec.gif",fps=8,show_msg=false)

The circular motion

• Define the trajectory function

rx_c(t)=r.*cos.(ω.*t);
ry_c(t)=r.*sin.(ω.*t);

r=2;ω=π/6;

t=0:0.1:12;plot(rx_c(t),ry_c(t),lab="circular",linewidth=2,color=:red,xlims=(-3,3),ylims=(-2.1,2.1))

p=plot(rx_c,ry_c,zeros(1),lab="circular",linewidth=2,color=:red,xlims=(-3,3),ylims=(-2.1,2.1))
circ=@animate for t=0:0.1:12
    push!(p,rx_c(t),ry_c(t))
    end;

gif(circ,"circular.gif",fps=8,show_msg=false)