Although it was very controversial in the 17 th and 18 th century Europe, the practical aspects of using infinitesimal quantities in calculations led to advances in science, engineering, and technology, along with the development of calculus. A theorem on the theory of quotient (Boolean) algebras follows from these results.Using infinitesimal quantities to approximate measurement of any item is an ancient way to determine the size and shape of irregular objects. Theorems 2 and 3 hold for more general $(*). There does not exist a one-to-one mapping f from I X I onto itself, such that K E $(*) iff f(K) is a Lebesgue measure zero (first category) subset of I x I. $(*) does not satisfy (vi) There is a subclass T of power c of the class $(^) such that every member of the class is contained in some member of the subclass. (v) The unit square may be represented as the union of two complementary Borel sets: one in $ and the other one in *. as the union of two complementary Borel sets: one in $ andit and the other one of Lebesgue measure zero and first category. (iii) The complement of each member of $ or V contains a set of power c belonging to $ and V, respectively. Let $(*) denote the family of subsets of the unit square defined to be of first category (Lebesgue measure zero) in almost every vertical line in the sense of measure (category). A theorem on the theory of quotient (Boolean) algebras follows from these results. Theorems 2 and 3 hold for more general $\Phi(\Psi)$. There does not exist a one-to-one mapping $f$ from $I \times I$ onto itself, such that $K \in \Phi(\Psi) iff f(K)$ is a Lebesgue measure zero (first category) subset of $I \times I$. $\Phi(\Psi)$ does not satisfy (vi) There is a subclass $\Upsilon$ of power $\leqslant c$ of the class $\Phi(\Psi)$ such that every member of the class is contained in some member of the subclass. (v) The unit square may be represented as the union of two complementary Borel sets: one in $\Phi$ and the other one in $\Psi$. (iv) The unit square may be represented as the union of two complementary Borel sets: one in $\Phi$ and $\Psi$ and the other one of Lebesgue measure zero and first category. (iii) The complement of each member of $\Phi$ or $\Psi$ contains a set of power $c$ belonging to $\Phi$. (ii) The union of $\Phi$ or $\Psi$ is $I \times I$. (i) $\Phi$ and $\Psi$ are $\sigma$-ideals. Let $\Phi(\Psi)$ denote the family of subsets of the unit square defined to be of first category (Lebesgue measure zero) in almost every vertical line in the sense of measure (category). However, for us to achieve these results, it will be necessary to show that given d 1 ,d 2 ,d 3 ,⋯,d k ∈D then d 1 ,+d 2 +d 3 +⋯+d k =δ k ∈D k.
We then use this polynomial to define the rth differential of a function/using infinitesimals. In the results, we present a Higher Taylor’s formula in several variables using the defined infinitesimals. This step is necessary as it sets the basis for proving Higher Taylor’s formula in several variables. Then we introduce higher order infinitesimals, which are used to define higher order derivatives. First we introduce infinitestimal of square zero, which we use to define derivative and hence to present the Taylor’s formula in one variable. This paper introduces the infinitesimal approach to differentiation and then uses it to present a version of Higher Taylor’s polynomial in several variables.
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