Nonstandard Treatment of Two Dimensional Taylor Series with Reminder Formulas

The aim of this paper is to establish some new two dimensional Taylor series formulas using some concepts of nonstandard analysis given by Robinson and axiomatized by Nelson Keyword:, nonstandard analysis, infinitely near, Taylor series. ليات ةلسلستمل ةيسايق ريغ ةجلاعم و ر دعبُلا ةيئانث عم غيص لا اب يق دمح نامثع ميهاربإ مولعلا ةيلك ةعماج نيدلا حلاص :ثحبلا ملاتسا خيرات 19 / 8 / 2007 :ثحبلا لوبق خيرات 4 / 11 / 2007 ملا ل خ ص ثــــحبلا اذه نم فدــهلا نإ غيص داجيإ وه ن نـريتمم ا دـلل روليات ةلسلستمل ةديدج ـلذا دــ ـجا ذــلا ـــسايهلا نــرا يـــرلحتلا يياــام يـــعم مادمتــسام Robinson عـــضا ا Nilson قطنم بولسأم . :ةيحاتفملا تاملكلا ،يسايق ريغ ليلحت ليات ةلسلستم ،يئاهنلا برق و .ر 1Introduction: Let f be a continuous function defined on a domain D and posses its derivatives up to order n in D , then the Taylor development of ( ) f x about o x with remainder form is given by:

R x x t f t dt n Through this paper we need the following nonstandard concepts: Every set or element defined in a classical mathematics is called standard [1].

Definition 1.1
A real number x is called limited if there exists a positive standard real number r such that  x r , otherwise it is called unlimited. The set of all unlimited real numbers is denoted byR [1].

Definition 1.2
A real number x is called infinitesimal if  x r , for all positive standard real numbers r [1]

Definition 1.3
Two real numbers, x and y are infinitely close if − xy is infinitesimal, and is denoted by  xy [1].

2-Higher Order Differentiation
In [2] and [6] a brief introduction of higher order differentiation is given. Suppose that = ( , ) z f x y is a function of two variables with continuous partial derivatives of first order, then the differentiation of z , denoted by dz , is defined by: x y df x y f x y dx f x y dy , since dz is also a function of x and y , so if the second order partial derivatives of f exists then differentiation of dz exists, and it is called second order differentiation, which is denoted by 2 dz .
It is important to emphasize that the quantities dx and dy are assumed to be constants. Therefore we have: 3 .
x y x xx y yy xy xy

Proof:
Use mathematical induction to get the result.

3-Taylor Expansion of f(x,y)
Let f be a real valued function defined on a domainD , then The formulas (3.4) and (3.5) represent differential formulas of a Taylor series expansion with remainder. Similarly with a necessary modification we can define a Taylor series expansion of multiple variable functions [2], [5]. Let = ( , ) z f x y be a function of two variables defined in a rectangular region D such that its n -partial derivatives are defined and continuous in D . By using (3.1) and (3.4) we find that: cy [6].
Consequentially, with the first formula of (2.2) we can write the exponential Taylor expansion formula of a function of two variables as: In the next section we try to deduce new formulas of Taylor series with different forms of remainders.

4-Integral Formula of Taylor Series with Remainders
The integral formula of Taylor series of a function of two variables is based on the line integral on a curve. Let = ( , ) z f x y be a two variables function whose partial derivatives provided that the differentiation is not exact whenever we used it , since the line integral of exact differentiation will vanish.   f x y has a total differential of any order over a sectionally smooth curve C where C is a curve from     ( , ) f x y has a total differential of any order over a sectionally smooth curve C where C is a curve whose parametric equations are given by t . Therefore the Taylor series of f whose remainder is given by: