The Differential and Integral Calculi
- Formal Operations - See Math.44 Formal Calculus
- Topologically - Here
- Calculus::=Net{Differential_Calculus, Integral_Calculus, Ordinary_differential_equations, Partial_differential_equations,... }.
- Differential_Calculus::= Net{
- Classical_derivative::= Net{
- C::=(continuous)Real->Real,
- D::C<>->C,
- 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 given fuctions where the limit
only exists for certain values of x.
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- (above)|-D (_^n) = n * (_^(n-1)).
- (above)|-D(map[x](x^n)) = map[x](n*(x^(n-1))).
- (above)|-D sin = -cos, ... .
- (above)|-D ln = (1/_).
- (above)|-D (e^_) = (e^_).
- (above)|-D( f(g)) = D(f)(g) * D g.
- (above)|-D(u+v) = D u + D v.
- (above)|-D(u-v) = D u - D v.
- (above)|-D(c*u) = c*D(u), where c is a constant.
- (above)|-D(u*v) = u * (D v) + (D u) * v.
- (above)|-D(u/v) =
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- 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
- (above)|-D(e^_)=e^_.
However this is not the only solution because for constant a - (above)|-D(a*e^_) = a*e^_.
There is a rich, complex and elegant theory of these equations. These links [ ordinaryDiffEq.html ] [ ODE_resources.htm ] seem to be useful starting points.
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}=::Ordinary_differential_equations.For h: Real E^h::(C->C)=(map[f](map[x](f(x+h))).
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develop theory of expressions that treat E and D as an algebra: Operator_algebra(Real, {D,E}). - 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.
- (above)|- (Taylor): E^h = e^(h*D).
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}=::Classical_derivative. - Frechet_deriative::=
}=::Differential_Calculus. - Classical_derivative::=
- Integral_Calculus::= Net{
-
Here are two excellent resources:
[ Integral ]
to get you started. This page
[ Lists_of_integrals ]
has more integrals than I've seen in some books.
- Indefinite_integral::= Net{
- (above)|-D(f+const) = D(f).
- ∫egral_sign::=/D.
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An indefinite integral is indefinite because D is not (1)-(1):
}=::Indefinite_integral. - (above)|-D(f+const) = D(f).
- Definite_integral::= Net{
- ∫egral_sign:Closed_set(Number)->Integrable_function(Number,Number)->Number.
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}=::Definite_integral. - ∫egral_sign:Closed_set(Number)->Integrable_function(Number,Number)->Number.
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Integrals are, in general, harder to calculate than differentials:-(
}=::Integral_Calculus. - Indefinite_integral::=
- Partial_differential_equations::=
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 [ http://en.wikipedia.org/wiki/ ] (for example).
We can approach the calculus in two ways:
. . . . . . . . . ( end of section The Differential and Integral Calculi) <<Contents | End>>
Notes on MATHS Notation
Special characters are defined in
[ intro_characters.html ]
that also outlines the syntax of expressions and a document.
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
The notation used here is a formal language with syntax
and a semantics described using traditional formal logic
[ logic_0_Intro.html ]
plus sets, functions, relations, and other mathematical extensions.
For a more rigorous description of the standard notations see
Glossary