1. Calculus::=Net{Differential_Calculus, Operational_Calculus, Ordinary_differential_equations, Partial_differential_equations,...} .

2. Differential_Calculus::=
   Net{

   R::semi-ring. F::@(R->R). D::F<>->F. DF::=pre(D).
   1. For x,y,z:R, f, g, h :DF. ...
   2. For x:variable(R), e:expression(R), D[x](e) ::=D(map[x:R](e)).

      Thurston 94, Hugh Thurston, What is Wrong With the Definition of dy/dx, Am Math Monthly V101n9(Nov 94)pp855-857

   3. For f,g, d f/d g::F = D(f)/D(G).

      Convention: the variables x,y, and z are used to represent elements of F and in particular, x is often used as the Id function.

      
      

   4. |-Id in DF and D Id = 1.
      
   5. (above)|-D[x](x) = d x/d x=fun[x]1.
      
   6. |-For a:R, D[x](a)=[x]0.
      
   7. (above)|-D[x](a) = d a/d x=fun[x]0.
      
   8. |-D(f + g)= D(f) + D(g),
      
   9. |-D(f * g)= D(f) * g + f * D(g),
      
   10. |-D(f o g)= D(g) * (D(f) o g).
       
   11. (above)|-D(f ; g)=D(f) * (f ; D(g)).

       
       

   12. (above)|-For a,b:R, D(a * f + c)=a * D(f).

       
       

   13. (above)|-D[x](x^2)=fun[x]2*x, |-D[x](x^3)=fun[x]3*x^2, ... .
       
   14. (above)|-For n , D(_^n)=n*(_^(n-1)).
       
   15. (above)|-For non_zero_integer n , D[x](x^n)=fun[x]n*x^(n-1).

       
       

   16. (above)|-D[x]f(g(x))=fun[x] ( D(g)(x) * (D(f))(g(x))).

       
       

   17. (above)|-if R in Field then D/f = 1/( D(f) o /f ).

   
   }=::Differential_Calculus.

3. Operational_Calculus::=
   Net{

   1. R::semi-ring.
   2. F::@(R->R).
   3. n::=dimension(R). Assume n different and non-overlapping step operators.
   4. D::F<>->F.
   5. DF::=pre(D).
   6. E::1..n->(F<>->F)= fun[i](fun[f](fun[r](f(r+δ[i])).
   7. Δ::= E - 1.
   8. Δ_upside_down::= 1 - /E.

      
      

   9. |-semi_ring( generated=>Net{D, F, Δ, Δ_upside_down, ...}.

      
      

   10. (above)|-if n=1, ... then E= e^(h D).

   
   }=::Operational_Calculus.

. . . . . . . . . ( end of section Calculus) <<Contents | End>>

Notes on MATHS Notation

Proofs follow a natural deduction style that start with assumptions ("Let") and continue to a consequence ("Close Let") and then discard the assumptions and deduce a conclusion. Look here [ Block Structure in logic_25_Proofs ] for more on the structure and rules.

The notation also allows you to create a new network of variables and constraints. A "Net" has a number of variables (including none) and a number of properties (including none) that connect variables. You can give them a name and then reuse them. The schema, formal system, or an elementary piece of documentation starts with "Net" and finishes "End of Net". For more, see [ notn_13_Docn_Syntax.html ] for these ways of defining and reusing pieces of logic and algebra in your documents. A quick example: a circle = Net{radius:Positive Real, center:Point}.

For a complete listing of pages in this part of my site by topic see [ home.html ]

Notes on the Underlying Logic of MATHS

For a more rigorous description of the standard notations see