.Open Calculus Calculus::=Net{Differential_Calculus, Operational_Calculus, Ordinary_differential_equations, Partial_differential_equations,...} . Differential_Calculus::=Net{ R::semi-ring=given. F::@(R->R). D::F<>->F=given. DF::=pre(D), differentiable functions. For x,y,z:R, f, g, h :DF. ... For x:variable(R), e:expression(R), D[x](e) ::=D(map[x:R](e)). .Source Thurston 94, Hugh Thurston, What is Wrong With the Definition of dy/dx, Am Math Monthly V101n9(Nov 94)pp855-857 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. |- Id in DF and D Id = 1. ()|-D[x](x) = d x/d x=fun[x]1. |- For a:R, D[x](a)=[x]0. ()|-D[x](a) = d a/d x=fun[x]0. |- D(f + g)= D(f) + D(g), |- D(f * g)= D(f) * g + f * D(g), |- D(f o g)= D(g) * (D(f) o g). ()|-D(f ; g)=D(f) * (f ; D(g)). ()|-For a,b:R, D(a * f + c)=a * D(f). ()|-D[x](x^2)=fun[x]2*x, ()|-D[x](x^3)=fun[x]3*x^2, ... . ()|-For n , D(_^n)=n*(_^(n-1)). ()|-For non_zero_integer n , D[x](x^n)=fun[x]n*x^(n-1). ()|-D[x]f(g(x))=fun[x] ( D(g)(x) * (D(f))(g(x))). ()|-if R in Field then D/f = 1/( D(f) o /f ). }=::Differential_Calculus. Operational_Calculus::=Net{ R::semi-ring=given. F::@(R->R). n::=dimension(R)=given. Assume \$n different and non-overlapping step operators. D::F<>->F=given. DF::=pre(D). E::1..n->(F<>->F)= fun[i](fun[f](fun[r](f(r+\delta[i])). \Delta::= E - 1. \Delta_upside_down::= 1 - /E. |- semi_ring( generated=>Net{D, F, \Delta, \Delta_upside_down, ...}. ()|- if n=1, ... then E= e^(h D). }=::Operational_Calculus. .Close Calculus