Related papers: Calculus without Limit Theory
A simple but rigorous proof of the Fundamental Theorem of Calculus is given in geometric calculus, after the basis for this theory in geometric algebra has been explained. Various classical examples of this theorem, such as the Green's and…
In this work we look at the original fractional calculus of variations problem in a somewhat different way. As a simple consequence, we show that a fractional generalization of a classical problem has a solution without any restrictions on…
When teaching an elementary logic course to students who have a general scientific background but have never been exposed to logic, we have to face the problem that the notions of deduction rule and of derivation are completely new to them,…
We present the basic theory of calculus on dual real numbers, and prove the counterpart of the ordinary fundamental theorem of calculus in the context of dual real numbers.
This is an introduction to calculus, and its applications to basic questions from physics. We first discuss the theory of functions $f:\mathbb R\to\mathbb R$, with the notion of continuity, and the construction of the derivative $f'(x)$ and…
We generalise the Fundamental Theorem of Calculus to higher dimensions. Our generalisation is based on the observation that the antiderivative of a function of $n$-variables is a solution of a partial differential equation of order $n$…
Projectile motion is a constant theme in introductory-physics courses. It is often used to illustrate the application of differential and integral calculus. While most of the problems used for this purpose, such as maximizing the range, are…
There is a deformation of the ordinary differential calculus which leads from the continuum to a lattice (and induces a corresponding deformation of physical theories). We recall some of its features and relate it to a general framework of…
I explain a direct approach to differentiation and integration. Instead of relying on the general notions of real numbers, limits and continuity, we treat functions as the primary objects of our theory, and view differentiation as division…
In this paper, we investigate bounded action theories in the situation calculus. A bounded action theory is one which entails that, in every situation, the number of object tuples in the extension of fluents is bounded by a given constant,…
Let $\mathcal{A}$ denote a real, $n$-dimensional, unital, associative algebra.This paper provides an introductory exposition of calculus over $\mathcal{A}$. An $\mathcal{A}$-differentiable function is one for which the differential is…
Most of theoretical physics is based on the mathematics of functions of a real or a complex variable; yet we frequently are drawn to try extending our reach to include quaternions. The non-commutativity of the quaternion algebra poses…
Universal geometric calculus simplifies and unifies the structure and notation of mathematics for all of science and engineering, and for technological applications. This paper treats the fundamentals of the multivector differential…
In a previous article we gave the general foundations of the theory of movement considered from a philosophical and mathematical point of view. Philosophical it meant to understand the opposition of the one and the multiple, mathematically…
This work introduces a new approach to velocity averaging lemmas in kinetic theory. This approach -- based upon the classical energy method -- provides a powerful duality principle in kinetic transport equations which allows for a natural…
A major question in philosophy of science involves the unreasonable effectiveness of mathematics in physics. Why should mathematics, created or discovered, with nothing empirical in mind be so perfectly suited to describe the laws of the…
In this paper, we study the consequences of the fundamental theorem of calculus from an algebraic point of view. For functions with singularities, this leads to a generalized notion of evaluation. We investigate properties of such…
In this article we discuss an important students' misconception about derivatives, that the expression of the derivative of the function contains the information as to whether the function is differentiable or not where the expression is…
H\"older functions represent mathematical models of nonlinear physical phenomena. This work investigates the general conditions of existence of fractional velocity as a localized generalization of ordinary derivative with regard to the…
A new derivative, called deformable derivative, is introduced here which is equivalent to ordinary derivative in the sense that one implies other. The deformable derivative is defined using limit approach like that of ordinary one but with…