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(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
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Models for Smooth Infinitesimal Analysis
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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.
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 $A:\mathbb{R} \to \langle \mathbb{G}^n \rangle_2$
where $a \wedge b$ is the wedge product of two vectors $a$ and $b$, and $\overrightarrow{v}$ is the velocity.
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
with the areal velocity:
Conservation of areal velocity is the same as the conservation of angular momentum.
See also:
Last revised on May 17, 2022 at 15:38:25. See the history of this page for a list of all contributions to it.