nLab areal velocity




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id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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 }


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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 nn-dimensional Euclidean space n\mathbb{R}^n, one could select an orthonormal basis on n\mathbb{R}^n by postulating an origin 00 and a function i^:[1,n] n\hat{i}:[1, n] \to \mathbb{R}^n such that for all m,p[1,n]m, p \in [1, n] the pair of vectors i^ m\hat{i}_m and i^ p\hat{i}_p is mutually orthonormal. There is an geometric algebra 𝔾 n\mathbb{G}^n on n\mathbb{R}^n defined by the equations i^ m 2=1\hat{i}_m^2 = 1 for all m[1,n]m \in [1, n], and i^ mi^ p=i^ pi^ m\hat{i}_m \hat{i}_p = -\hat{i}_p \hat{i}_m for all m,p[1,n]m, p \in [1, n].

A smooth curve 𝒞\mathcal{C} in n\mathbb{R}^n could be parameterized by a smooth function r: n\overrightarrow{r}:\mathbb{R} \to \mathbb{R}^n. Then the areal velocity, sector velocity, or sectorial velocity of a point in 𝒞\mathcal{C} in n\mathbb{R}^n is given by the bivector-valued function A:𝔾 n 2A:\mathbb{R} \to \langle \mathbb{G}^n \rangle_2

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

where aba \wedge b is the wedge product of two vectors aa and bb, and v\overrightarrow{v} is the velocity.

In 3 dimensions

In 3 dimensions, the vector areal velocity a:𝔾 n 1\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= i:[1,n]i^ iI = \prod_{i:[1, n]} \hat{i}_i

with the areal velocity:

a(t)=IA(t)=I(r(t)v(t)2)=I(r(t)v(t))2=r(t)×v(t)2\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.

See also


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.