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The Fundamental Theorem of Calculus in higher dimensions

Filip Bár

Abstract

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 $n$ generalising the classical case. The generalised Fundamental Theorem of Calculus then states that the $n$-dimensional integrals over $n$-dimensional axis-parallel rectangular hypercuboids is given by a combinatorial formula evaluating the antiderivative on the vertices of the hypercuboid.

The Fundamental Theorem of Calculus in higher dimensions

Abstract

We generalise the Fundamental Theorem of Calculus to higher dimensions. Our generalisation is based on the observation that the antiderivative of a function of -variables is a solution of a partial differential equation of order generalising the classical case. The generalised Fundamental Theorem of Calculus then states that the -dimensional integrals over -dimensional axis-parallel rectangular hypercuboids is given by a combinatorial formula evaluating the antiderivative on the vertices of the hypercuboid.
Paper Structure (3 sections, 1 theorem, 17 equations)

This paper contains 3 sections, 1 theorem, 17 equations.

Table of Contents

  1. Introduction
  2. The Fundamental Theorem of Calculus
  3. Conclusion and Outlook

Key Result

Theorem 2.1

Let $I=\prod_{j=1}^{n} [a_j,b_j]$, $\mathring{I}=\prod_{j=1}^{n} (a_j,b_j)$ and $f:I\to \mathbb{R}$ a continuous function.

Theorems & Definitions (2)

  • Theorem 2.1: Fundamental Theorem of Calculus
  • proof