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Thu Feb 21 21:08:31 PST 2013

# 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 [ http://en.wikipedia.org/wiki/ ] (for example).

We can approach the calculus in two ways:

1. Formal Operations - See Math.44 Formal Calculus
2. Topologically - Here

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

## Differential Calculus

2. Differential_Calculus::=
Net{

1. Classical_derivative::=
Net{
1. C::=(continuous)Real->Real,
2. D::C<>->C, only some functions can be differentiated.
3. differentiable::@C=cor(D),
4. |-for f:C, D f= map[x:Real]lim[h+>0]((f(x+h)-f(x))/h).

5. 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.

6. (above)|-D (_^n) = n * (_^(n-1)).
7. (-1)|-D(map[x](x^n)) = map[x](n*(x^(n-1))).
8. (above)|-D sin = cos, D cos = -sin, D tan = sec^2, ... .
9. (above)|-D ln = (1/_).
10. (above)|-D (e^_) = (e^_).
11. (above)|-D( f(g)) = D(f)(g) * D g.
12. (above)|-D(u+v) = D u + D v.
13. (above)|-D(u-v) = D u - D v.
14. (above)|-D(c*u) = c*D(u), where c is a Real constant.
15. (above)|-D(u*v) = u * (D v) + (D u) * v.
16. (above)|-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.

[click here D (_)^(_) = ?. if you can fill this hole]

17. For f,g:differentiable, d f/ d g::=D(f)/D(g).

18. The classic Liebnitz notation: dy/dx stands (roughly) for
19. (D map[x]y)(x).

20. Ordinary_differential_equations::=
Net{
1. 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.

2. For example the equation
3. (ode1): D f = f.
4. has a solution because
5. (above)|-D(e^_)=e^_.
6. However this is not the only solution because for constant a
7. (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.

}=::Ordinary_differential_equations.

For h: Real E^h::(C->C)=(map[f](map[x](f(x+h))).

21. develop theory of expressions that treat E and D as an algebra: Operator_algebra(Real, {D,E}).
22. (above)|- (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:

23. For n:Nat, C(n)::=(continuous)Real->(Real^n),
24. For f:C(n), define a vector of n functions f[i]:Real->Real,
25. f(t) = (f1(t), f2(t),... f[n](t)),
26. D::C(n)<>->C(n)= map[f:C(n)]map[t:Real](map[i:1..n](D(f[i](t))).

}=::Classical_derivative.

2. Frechet_deriative::=
Net{

}=::Frechet_derivative.

## Partial Differentials

In a formula like
3. 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.

4. ∂ y / ∂ x.

The MATHS symbol for the curly d of classical mathematics is the same as in ΤΕΧ:

 		\partial y / \partial x

It means that y equals an expression and all the variables in it are assumed to be constants except x.

}=::Differential_Calculus.

## Integral Calculus

3. 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.

Integrals are, in general, harder to calculate than differentials:-( There is no efficient algorithm, like that for differentiation, for integrating a function.

1. Indefinite_integral::=
Net{
1. An indefinite integral is indefinite because D is not (1)-(1):
2. (above)|-for all f:differentiable, c:Real, D(f+c) = D(f).

3. Const::= { f:C || for some c:Real ( f=map[x:Real](c)), the set of const functions.

4. ::=/D.
5. integral_sign::=∫.
 			\int

6. (above)|-∫(2*_) = (_)^2 + Const.

}=::Indefinite_integral.

2. Definite_integral::=
Net{
A definite integral is given a integrable function and a closed set to integrate over, returns a number.
1. ::Integrable_function->(Closed_set(Number)->Number).
2. Integrable_function::@(Number->Number).

3. (above)|-∫(2*_)[a..b] = b^2 - a^2.

}=::Definite_integral.

}=::Integral_Calculus.

4. Partial_differential_equations::=
Net{

}=::Partial_differential_equations.

. . . . . . . . . ( 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 might be described by Net{radius:Positive Real, center:Point, area:=π*radius^2, ...}.

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

1. STANDARD::= See http://www.csci.csusb.edu/dick/maths/math_11_STANDARD.html

# Glossary

2. above::reason="I'm too lazy to work out which of the above statements I need here", often the last 3 or 4 statements. The previous and previous but one statments are shown as (-1) and (-2).
3. given::reason="I've been told that...", used to describe a problem.
4. given::variable="I'll be given a value or object like this...", used to describe a problem.
5. goal::theorem="The result I'm trying to prove right now".
6. goal::variable="The value or object I'm trying to find or construct".
7. let::reason="For the sake of argument let...", introduces a temporary hypothesis that survives until the end of the surrounding "Let...Close.Let" block or Case.
8. hyp::reason="I assumed this in my last Let/Case/Po/...".
9. QED::conclusion="Quite Easily Done" or "Quod Erat Demonstrandum", indicates that you have proved what you wanted to prove.
10. QEF::conclusion="Quite Easily Faked", -- indicate that you have proved that the object you constructed fitted the goal you were given.
11. RAA::conclusion="Reducto Ad Absurdum". This allows you to discard the last assumption (let) that you introduced.