{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/representing-and-computing-the-b-derivative","title":"Representing and computing the B-derivative of an $EC^r$ vector field's $PC^r$ flow","arxiv_id":"2102.10702","date":"2021-02-21","proceeding":null,"authors":["George Council","Shai Revzen","Samuel A. Burden"],"abstract":"This paper concerns the first-order approximation of the piecewise-differentiable flow generated by a class of nonsmooth vector fields. Specifically, we represent and compute the Bouligand (or B-)derivative of the piecewise-$C^r$ flow generated by an event-selected $C^r$ vector field. Our results are remarkably efficient: although there are factorially many \"pieces\" of the desired derivative, we provide an algorithm that evaluates its action on a given tangent vector using polynomial time and space, and verify the algorithm's correctness by deriving a representation for the B-derivative that requires \"only\" exponential time and space to construct. We apply our methods in two classes of illustrative examples: piecewise-constant vector fields and mechanical systems subject to unilateral constraints.","url_abs":"https://arxiv.org/abs/2102.10702v1","url_pdf":"https://arxiv.org/pdf/2102.10702v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"representing-and-computing-the-b-derivative","repo_url":"https://github.com/gcouncil-cmu/bderv","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}