# nLab areal velocity

Contents

## Surveys, textbooks and lecture notes

Ingredients

Concepts

Constructions

Examples

Theorems

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

#### Variational calculus

variational calculus

# Contents

## Idea

The bivector whose magnitude equals the rate of change at which area is swept out by a point/particle as it moves along a curve.

## Definition

Given an $n$-dimensional Euclidean space $\mathbb{R}^n$, one could select an orthonormal basis on $\mathbb{R}^n$ by postulating an origin $0$ and a function $\hat{i}:[1, n] \to \mathbb{R}^n$ such that for all $m, p \in [1, n]$ the pair of vectors $\hat{i}_m$ and $\hat{i}_p$ is mutually orthonormal. There is an geometric algebra $\mathbb{G}^n$ on $\mathbb{R}^n$ defined by the equations $\hat{i}_m^2 = 1$ for all $m \in [1, n]$, and $\hat{i}_m \hat{i}_p = -\hat{i}_p \hat{i}_m$ for all $m, p \in [1, n]$.

A smooth curve $\mathcal{C}$ in $\mathbb{R}^n$ could be parameterized by a smooth function $\overrightarrow{r}:\mathbb{R} \to \mathbb{R}^n$. Then the areal velocity, sector velocity, or sectorial velocity of a point in $\mathcal{C}$ in $\mathbb{R}^n$ is given by the bivector-valued function $A:\mathbb{R} \to \langle \mathbb{G}^n \rangle_2$

$A(t) = \frac{\overrightarrow{r}(t) \wedge \overrightarrow{v}(t)}{2}$

where $a \wedge b$ is the wedge product of two vectors $a$ and $b$, and $\overrightarrow{v}$ is the velocity.

## In 3 dimensions

In 3 dimensions, the vector areal velocity $\overrightarrow{a}:\mathbb{R} \to \langle \mathbb{G}^n \rangle_1$ is the Hodge dual of the areal velocity, which is the product of the pseudoscalar

$I = \prod_{i:[1, n]} \hat{i}_i$

with the areal velocity:

$\overrightarrow{a}(t) = I A(t) = I\left(\frac{\overrightarrow{r}(t) \wedge \overrightarrow{v}(t)}{2}\right) = \frac{I(\overrightarrow{r}(t) \wedge \overrightarrow{v}(t))}{2} = \frac{\overrightarrow{r}(t) \times \overrightarrow{v}(t)}{2}$

## Conservation of areal velocity

Conservation of areal velocity is the same as the conservation of angular momentum.