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Nonholonomic Behavior in Robotic Systems
Published in Richard M. Murray, Zexiang Li, S. Shankar Sastry, A Mathematical Introduction to Robotic Manipulation, 2017
Richard M. Murray, Zexiang Li, S. Shankar Sastry
Associated with the tangent space Tqℝn is the dual space Tq*ℝn, the set of linear functions on Tqℝn. Just as we defined vector fields on ℝn, we define a one-form as a map which assigns to each point q ∈ ℝn a convectorω(q)∈Tq*ℝn. In local coordinates we represent a smooth one-form as a row vector ω(q)=[ω1(q)ω2(q)⋯ωn(q)].Differentials of smooth functions are good examples of one-forms. For example, if β : ℝn → ℝ, then the one-form dβ is given by dβ=[∂β∂q1∂β∂q2⋯∂β∂qn]. Note, however, that all one-forms are not necessarily the differentials of smooth functions (a one-form which does happen to be the derivative of a function is said to be exact).
On the Symmetry of Blast Waves
Published in Nuclear Technology, 2021
which may be used to analyze the local geometric properties of Eq. (11). From a classical perspective, the expression of Eq. (12) represents the infinitesimal change of under an infinitesimal change of the point on the underlying geometric object where the expression is evaluated. Since geometric infinitesimals are not rigorously defined and because total derivatives may be defined along tangent vectors, is interpreted as a differential one-form (see Spivak25). Differential one-forms are the objects dual to vector fields defined locally on curves and surfaces. The use of differential forms is very powerful in the analysis of systems of PDEs. By admitting higher-order exterior forms, smooth systems of PDEs are equivalent to exterior differential systems (EDSs), which may be analyzed with mathematical techniques from differential and algebraic geometry, where an EDS is a set of equations in terms of differential forms (see Krasil’shchik and Vinogradov26 and Vinogradov27).