.Open The Differential and Integral Calculi There is a lot of work to be done here. However classical mathematics has a 200 year history of developments that are already well documented on the world wide web .See http://en.wikipedia.org/wiki/ (for example). We can approach the calculus in two ways: .List Formal Operations - See Math.44 Formal Calculus Topologically - Here .Close.List Calculus::=Net{$Differential_Calculus,$Integral_Calculus, $Ordinary_differential_equations,$Partial_differential_equations,... }. . Differential Calculus Differential_Calculus::=Net{ Classical_derivative::=Net{ C::=(continuous)Real->Real, D::C<>->C, only some functions can be differentiated. differentiable::@C=cor(D), |-for f:C, D f= map[x:Real]lim[h+>0]((f(x+h)-f(x))/h). Differentiable functions are functions where the limit above exists and is unique. A more complicated version would allow the D operator to produce partial functions when applied to functions where the limit only exists for certain values of x. .Hole ()|- D (_^n) = n * (_^(n-1)). (-1)|- D(map[x](x^n)) = map[x](n*(x^(n-1))). ()|- D sin = cos, D cos = -sin, D tan = sec^2, ... . ()|- D ln = (1/_). ()|- D (e^_) = (e^_). ()|- D( f(g)) = D(f)(g) * D g. ()|- D(u+v) = D u + D v. ()|- D(u-v) = D u - D v. ()|- D(c*u) = c*D(u), where c is a Real constant. ()|- D(u*v) = u * (D v) + (D u) * v. ()|- D(u/v) = ( (D u) * v - u*(D v) )/(v^2). Notice that we can apply the above rules to any elementary expression, top-down to calculate the 'D' of the expression with respect to any one variable. .Hole D (_)^(_) = ?. For f,g:differentiable, d f/ d g ::=D(f)/D(g). The classic Liebnitz notation: dy/dx stands (roughly) for (D map[x]y)(x). Ordinary_differential_equations::=Net{ Ordinary differential equations are a collection of equations that include several variables and their classical derivatives. They have proved a powerful tool for modeling life, death and the universe. For example the equation (ode1): D f = f. has a solution because ()|- D(e^_)=e^_. However this is not the only solution because for constant a ()|- D(a*e^_) = a*e^_. There is a rich, complex and elegant theory of these equations. These links .See http://archives.math.utk.edu/topics/ordinaryDiffEq.html .See http://www.math.tamu.edu/~Don.Allen/ODE_resources.htm seem to be useful starting points. .Hole }=::Ordinary_differential_equations. For h: Real E^h::(C->C)=(map[f](map[x](f(x+h))). .Hole develop theory of expressions that treat E and D as an algebra: Operator_algebra(Real, {D,E}). ()|-(Taylor): E^h = e^(h*D). We can easily extend the definition of the derivative D to functions mapping real numbers into n-tuples of reals by treating them as n differentiable functions: For n:Nat, C(n)::=(continuous)Real->(Real^n), For f:C(n), define a vector of n functions f[i]:Real->Real, f(t) = (f1(t), f2(t),... f[n](t)), D::C(n)<>->C(n)= map[f:C(n)]map[t:Real](map[i:1..n](D(f[i](t))). .Hole }=::Classical_derivative. Frechet_deriative::=Net{ .Hole }=::Frechet_derivative. . Partial Differentials In a formula like d ( y^2 + x^2 )/d x in classical mathematics some of the variables represent functions of a dependent variable and other symbols are constants. Typically, x is the independent variable and y, z, w, u, v, ... are assumed to be functions of x -- normally. However it is useful to be able to make all but one of the variables constants. The so-called partial differentials. There are several theorems that express a normal differential as the sum of terms where each term is made of a differential and a partial differential, for example. On the applied side many physical laws can be expressed as partial differential equation. This a partial differential. \partial y / \partial x. The MATHS symbol for the curly d of classical mathematics is the same as in \TeX: .As_is \partial y / \partial x It means that y equals an expression and all the variables in it are assumed to be constants except x. .Hole }=::Differential_Calculus. . Integral Calculus Integral_Calculus::=Net{ Here are two excellent resources: .See http://en.wikipedia.org/wiki/Integral to get you started. This page .See http://en.wikipedia.org/wiki/Lists_of_integrals has more integrals than I've seen in some books. Integrals are, in general, harder to calculate than differentials:-( There is no efficient algorithm, like that for differentiation, for integrating a function. Indefinite_integral::= Net{ An indefinite integral is indefinite because D is not (1)-(1): ()|- for all f:differentiable, c:Real, D(f+c) = D(f). Const::= { f:C || for some c:Real ( f=map[x:Real](c)), the set of const functions. \int::=/D. integral_sign::=\int. .As_is \int ()|- \int(2*_) = (_)^2 + Const. .Hole }=::Indefinite_integral. Definite_integral::= Net{ A definite integral is given a integrable function and a closed set to integrate over, returns a number. \int::Integrable_function->(Closed_set(Number)->Number). Integrable_function::@(Number->Number). .Hole ()|- \int(2*_)[a..b] = b^2 - a^2. .Hole }=::Definite_integral. }=::Integral_Calculus. Partial_differential_equations::=Net{ .Hole }=::Partial_differential_equations. .Close The Differential and Integral Calculi