Work Done Calculation to Rotate the Disk

1. • -- multiply
   
2. * -- multiply but may be other
   
3. == -- same as
   
4. ^ or ** -- power (but dominantly superscript) a^p == a**p == a^p == exp(p•ln(a))
   

1. ( a ± b )² = a² ± 2ab + b²
   
   
2. ( a ± b )³ = a³ ± 3a²b + 3ab² ± b³
   
   
3. ( a - b )( a + b ) = a² - b²
   ( a - ib )( a + ib ) = a² + b²
   
   
4. ( a ± b )( a² ± (-ab) + b² ) = a³ ± b³
   
   

• Sum_{[ c = (0,)1..n ]} ( 2c ) = n( n + 1 ) = (0 +)2 + 4 + 6 + 8 + .. + 2n
  
  
• Sum_{[ c = 1..n ]} ( 2c-1 ) = n² = 1 + 3 + 5 + 7 + 9 + .. + ( 2n - 1 )
  
  
• Sum_{[ c = (0,)1..n ]} ( c ) = n( n + 1 )/2 = (0 +)1 + 2 + 3 + 4 + 5 + .. + n
  
  
• Sum_{[ c = 1..n ]} ( c² ) = n( n + 1 )•( 2n + 1 )/6 = 1 + 4 + 9 + 16 + 25 + .. + n²
  
  
• Sum_{[ c = 1..n ]} ( c³ ) = [n( n + 1)/2]² = 1 + 8 + 27 + 64 + 125 + .. + n³
  
  
• Sum_{[ c = 0..n ]} ( a^c ) = ( a^{( n + 1 )} - 1 )/( a - 1 ) = 1 + a + a² + a³ + a⁴ + .. + aⁿ
  
  

• A_n=A₁+(n-1)d -- (0=A₁),2,4,6,(8=A₅), d=2
  
  
• S_n=n(A₁+A_n)/2=(2A₁+(n-1)d)n/2 -- 0+2+4+6+8=5*(0+8)/2=(20+(5-1)2)5/2=20
  
  
• S_n3=(A_n2+A_n1+(n₃-n₂-n₁+1)(A_n2-A_n1):(n₂-n₁)n₃/2 -- 0+2+4+6=(2+0+(4-2-1+1)(2-0)/(2-1))4/2=12
  
  

• A_n=A₁q^n-1 -- (1=A₁),2,4,8,(16=A₅), q=2
  
  
• S_n=A₁(q_n-1)/(q-1) -- 1+2+4+8+16=1*(2⁵-1)/(2-1)=31
  
  
• S_{n->inf,q=abs(q)<1}=A₁/(q-1) -- 6/1+6/2+6/4+6/8+6/16+..+6/2^n,n->inf=6/(1-1/2)=12
  
  

1. Sin(a±b)=Sin(a)Cos(b)±Cos(a)Sin(b)
   
   
2. Cos(a±b)=Cos(a)Cos(b)±(-Sin(a)Sin(b))
   
   
3. Tan(a±b)=(Tan(a)±Cos(b)):(1±(-Tan(a)Sin(b)))
   
   
4. Cot(a±b)=(Cot(a)Cot(b)±(-1)):(Cot(a)±Cot(b))
   
   

1. Sin(a)±Sin(b)=2Sin((a±b):2)Cos((a±(-b)):2)
   
   
2. Cos(a)+Cos(b)=2Cos((a+b):2)Cos((a-b):2)
   
   Cos(a)-Cos(b)=-2Sin((a+b):2)Sin((a-b):2)
   
   
3. Tan(a)±Tan(b)=Sin(a±b):(Cos(a)Cos(b))
   
   
4. Cot(a)±Cot(b)=Sin(a±b):(Sin(a)Sin(b))
   
   

1. Cos²(a)+Sin²(a)=1=(Cos(a)-iSin(a))(Cos(a)+iSin(a))=e^i(-a)e^ia=(1/e^ia)e^ia
   
   
2. Tanⁿ(a)=Cot^-n(a)=(Sin(a)/Cos(a))ⁿ
   
   
   
3. Sin(2a)=2Sin(a)Cos(a)
   
   
4. Cos(2a)=Cos²(a)-Sin²(a)=2Cos²(a)-1=1-2Sin²(a)
   
   
5. Tan(2a)=(2Tan(a)):(1-Tan²(a))
   
   
6. Cot(2a)=(Cot²(a)-1):(2Cot(a))
   
   
   
7. 1-Cos(a)=2Sin²(a/2)
   
   
8. 1+Cos(a)=2Cos²(a/2)
   
   
9. 1-Sin(a)=(Cos(a/2)-Sin(a/2))²
   
   
10. 1+Sin(a)=(Cos(a/2)+Sin(a/2))²
    
    
    
11. Sin(a/2)=±((1-Cos(a)):2)^1/2
    
    
12. Cos(a/2)=±((1+Cos(a)):2)^1/2
    
    
13. Tan(a/2)=±((1-Cos(a)):(1+Cos(a)))^1/2=(1-Cos(a))/Sin(a)=Sin(a)/(1+Cos(a))
    
    
    
14. 1+Tan²(a)=1/Cos²(a)
    
    
15. 1+Cot²(a)=1/Sin²(a)
    
    
    
16. Sec(a)=1/Cos(a)=(1+Tan²(a))^1/2
    
    
17. Cosec(a)=1/Sin(a)=(1+Cot²(a))^1/2
    
    
    

1. arccos(A)=±iln(A±(A²-1)^1/2)
   
   
2. arcsin(A)=(±)iln(±(1-A²)^1/2-(±)iA), [±&(±)] are inipendent and [±&± or (±)&(±)] are syncronized
   
   
3. i/2ln((1-iA):(1+iA))=arctan(A)=i/2ln((i+A):(i-A))
   
   
4. i/2ln((iA+1):(iA-1))=arccot(A)=i/2ln((A-i):(A+i))
   
   
   
5. arctan(A)+arccot(A)=Pi/2±2nPi, n is ±integer or 0
   
   
6. ±(arctan(A)-arccot(A))=i/2ln(i(iA-±1):(iA±1))²=i/2ln(i(1-±iA):(1±iA))²
   
   

1. ch(a±b)=ch(a)sh(b)±sh(a)sh(b)
   
   
2. sh(a±b)=sh(a)ch(b)±ch(a)sh(b)
   
   
3. th(a±b)=(th(a)±th(b)):(1±th(a)th(b))
   
   
4. cth(a±b)=(1±cth(a)cth(b)):(cth(a)±cth(b))
   
   

1. ch(a)+ch(b)=2ch((a+b):2)ch((a-b):2)
   
   ch(a)-ch(b)=2sh((a+b):2)sh((a-b):2)
   
   
2. sh(a)±sh(b)=2sh((a±b):2)ch((a±(-b)):2)
   
   
3. th(a)±th(b)=sh(a±b):(ch(a)ch(b))
   
   
4. cth(a)±cth(b)=sh(a±b):(sh(a)sh(b))
   
   

1. ch²(a)-sh²(a)=1=(ch(a)-sh(a))(ch(a)+sh(a))=e^(-a)e^a=(1/e^a)e^a
   
   
2. thⁿ(a)=cth^-n(a)=(sh(a)/ch(a))ⁿ
   
   
   
3. ch(2a)=ch²(a)+sh²(a)=2ch²(a)-1=2sh²(a)+1
   
   
4. sh(2a)=2sh(a)ch(a)
   
   
5. th(2a)=(2th(a)):(1-th²(a))
   
   
6. cth(2a)=(1+cth²(a)):(2cth(a))
   
   
   
7. ch(a)-1=2ch²(a/2)
   
   
8. ch(a)+1=2sh²(a/2)
   
   
   
9. ch(a/2)=±((ch(a)+1):2)^1/2
   
   
10. sh(a/2)=±((ch(a)-1):2)^1/2
    
    
11. th(a/2)=±((ch(a)-1):(ch(a)+1))^1/2=(ch(a)-1)/sh(a)=sh(a)/(ch(a)+1)
    
    
    
12. 1-th²(a)=1/ch²(a)
    
    
13. 1-cth²(a)=-1/sh²(a)
    
    

1. arch(A)=ln(A±(A²-1)^1/2)
   
   
2. arsh(A)=ln(A±(A²+1)^1/2)
   
   
3. arth(A)=1/2ln((1+A):(1-A))
   
   
4. arcth(A)=1/2ln((A+1):(A-1))
   
   
   
5. arth(A)+arcth(A)=1/2ln(-(A+1):(A-1))²=1/2ln(-(1+A):(1-A))²
   
   
6. ±(arctan(A)-arccot(A))=ln(i)=i(Pi/2±2nPi), n is ±integer or 0
   
   

1. a^-x=1/a^x
   
   
2. F_m=0..3(x)=lim_n->+inf Sum_c=0..n(x^4c+m/(4c+m)!)
   
   
3. e^x=Sum_m=0..3(F_m(x))
   
   
4. ln(A)=ln((1+x₀)/(1-x₀))=lim_n->+inf 2!Sum_c=0..n(x₀^2c+1/(2c+1))
   
   
   

   F0(-x) = F0(x)	F0(ix) = 1F0(x)	F0(-ix) = 1F0(x)
   F1(-x) = -F1(x)	F1(ix) = iF1(x)	F1(-ix) = -iF1(x)
   F2(-x) = F2(x)	F2(ix) = -1F2(x)	F2(-ix) = -1F2(x)
   F3(-x) = -F3(x)	F3(ix) = -iF3(x)	F3(-ix) = iF3(x)

   
   
   
   

   F0(x)=[ch(x)+Cos(x)]/2	ch(x)=F0(x)+F2(x)=(ex+e-x)/2
   F1(x)=[ch(x)+Sin(x)]/2	sh(x)=F1(x)+F3(x)=(ex-e-x)/2
   F2(x)=[ch(x)-Cos(x)]/2	Cos(x)=F0(x)-F2(x)=(eix+e-ix)/2
   F3(x)=[sh(x)-Sin(x)]/2	Sin(x)=F1(x)-F3(x)=(eix-e-ix)/(2i)

   
   
   

1. log_x(ab)=log_x(a)+log_x(b)
   
   
2. log_x(a/b)=log_x(a)-log_x(b)
   
   
3. log_x(a^b)=blog_x(a)
   
   
4. log_x(a)=log_z(a)/log_z(x)=1/log_a(x)
   
   
   
5. z^1/ln(z)=e same as z=e^ln(z)
   
   
   
6. Operations famyly::
• ???,a$b=(area unknown requires 3D,4D numbers+ops. for simplex representation),a+b,ab,a*b,e.c.
  
  
• a*b=e^ln(a)ln(b) -- arithmetical xor operation
  
  
• a%b=e^ln(a)/ln(b) -- reverse xor-ing
  
  
• unitive property: for every next generation applies a op_n+1 b = e^{ln(a)op_nln(b)}
  
  
• Op. level base constants::
• -inf $ a = a = "u say" -- something close to ??? 1/(1/-inf+1/a) ???
• 0 + a = a
• 1 a = a
• e * a = a
• e^e (*"+1") a = a
• e.c.

1. z = a + ib = Re(z) + iIm(z) = |z|e^if = e^{ln|z| + iarg(z)}
   
   a = Re(z) -- real component of z,
   b = Im(z) -- imaginary component of z,
   |z| = abs[(a²+b²)^1/2] -- module of z
   f = arg(z) = arctan(b/a) -- argument of z in rad, 180°=Pi rad.
   
   graphical::
   
   
   
   
2. z = a - ib = Re(z) - iIm(z) = |z|e^-if = e^{ln|z| - iarg(z)} ,in futher refferences z =_def. @z
   
3. |z| = (z@z)^1/2 = |(Re²(z) + Im²(z))^1/2| = e.c. , can be fetched stright if exists f(z)=@z, f(@z)=z
   
   
4. arg(z) = arctan(Im(z)/Re(z)) = arctan(-i(z-@z)/(z+@z))
   
   

1. (u@u)(w@w)=(uw)@(uw)
   
   
2. |u||w|^±1=|(uw)^±1|
   
   
3. u + @u = 2Re(u)
   
   
4. u - @u = 2iIm(u)
   
   

1. zc^±1=e^{(ln|z|±ln|c|) + i(arg(z)±arg(c))}
   
   
2. zⁿ=e^{nln|z| + inarg(z)}
   
   z^1/n=e^{1/nln|z| + i/narg(z)}
   
   
1. z^w = e^wln(z) = e^{(Re(w)+iIm(w))(ln(z)+iarg(z))} =
   
   
2. = z^w = e^{(Re(w)ln|z|-Im(w)arg(z)) + i(Im(w)ln|z|+Re(w)arg(z))}
   
   
4. z=|z|e^if=|z|(cos(f)+iSin(f))=a+ib -- exp., trig., arit. - forms of z
   
   
5. e^if=F₀(if)+F₁(if)+F₂(if)+F₃(if)=
   
   =(F₀(f)-F₂(f))+i(F₁(f)-F₃(f))=Cos(f)+iSin(f)

e^x, ch(x), sh(x), Cos(x), Sin(x), F₀(x), F₁(x), F₂(x), F₃(x)

finest grid division's length = Pi/4