Analytic Geometry > In 3D space (also called xyz space), the xy plane contains the x-axis and y-axis: The xy plane can be described as the set of all points (x, y, z) where z = 0. In other words,…

## Weak Derivative: Simple Definition, Examples

Derivatives > The weak derivative uxi is an extension of the ordinary derivative to allow for the inclusion of problematic “points” that would otherwise make useful functions non-differentiable. For example, the absolute value function cannot be differentiated on the interval…

## Multiple Integrals: Definition, Examples

Integrals > Multiple integrals are definite integrals and are an extension of the ordinary (single) integral, which finds an area below a curve. Usually, the focus is on double integrals, which find the area of a region in a plane,…

## Ducci Sequence

Sequence and Series > A Ducci sequence (or N-number game) is a sequence of n-tuples (i.e. a list of integers). Each term in the sequence is formed by finding the absolute difference of neighboring integers [1]. Ducci Sequence Example Example…

## Absolutely Continuous

Types of functions > Absolutely continuous real-numbered functions are those functions for which the Fundamental Theorem of Calculus (FTC) holds [1]. In other words, absolute continuity identifies which functions can be antiderivatives: a function on a closed, bounded interval is…

## Umbral Calculus

Calculus Definitions > Umbral calculus (also called Blissard Calculus or Symbolic Calculus) is a modern way to do algebra on polynomials. It is a set of exploratory “rules” or a proof technique where indices of polynomial sequences are treated as…

## Ricci Calculus & Notation: Definition

Calculus Definitions > What is Ricci Calculus? Ricci calculus, sometimes called the calculus of congruences of curves [1], is the study of tensors and tensor fields. The modern name for Ricci calculus is tensor calculus, although the notation used within…

## Dirichlet Series: Overview, Notation

Sequence and Series > The Dirichlet series has the form [1]: Where: s is a real-valued [2] or complex variable (Dirichlet proposed that s should be real or complex, but Riemann later on stressed the importance of s as a…

## Pi Derivative & Derivative of Circle’s Area

Common Derivatives > Pi Derivative The derivative of π with respect to x is 0. π is a constant (equal to about 3.14), and the derivative of a constant is zero. Therefore π2, π/2, π/6, π/10 or any other π/constant…

## Limit Inferior and Limit Superior

Limit of Functions > The limit inferior and limit superior of a sequence give bounds on the sequence’s subsequential limits (i.e. the limit of a subsequence) [1]: The limit inferior is the smallest limit of a subsequence. The limit superior…