{"id":"https://openalex.org/W3210814943","doi":"https://doi.org/10.1137/19m1296732","title":"A Precise and Reliable Multivariable Chain Rule","display_name":"A Precise and Reliable Multivariable Chain Rule","publication_year":2021,"publication_date":"2021-01-01","ids":{"openalex":"https://openalex.org/W3210814943","doi":"https://doi.org/10.1137/19m1296732","mag":"3210814943"},"language":"en","primary_location":{"id":"doi:10.1137/19m1296732","is_oa":true,"landing_page_url":"https://doi.org/10.1137/19m1296732","pdf_url":"https://epubs.siam.org/doi/pdf/10.1137/19M1296732","source":{"id":"https://openalex.org/S160107561","display_name":"SIAM Review","issn_l":"0036-1445","issn":["0036-1445","1095-7200"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Review","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://epubs.siam.org/doi/pdf/10.1137/19M1296732","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5088147328","display_name":"Raymond Boute","orcid":"https://orcid.org/0000-0002-4329-3902"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Raymond Boute","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5088147328"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.5092,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.73856209,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":95},"biblio":{"volume":"63","issue":"4","first_page":"854","last_page":"864"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12044","display_name":"Advanced Physical and Chemical Molecular Interactions","score":0.9876000285148621,"subfield":{"id":"https://openalex.org/subfields/1606","display_name":"Physical and Theoretical Chemistry"},"field":{"id":"https://openalex.org/fields/16","display_name":"Chemistry"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12044","display_name":"Advanced Physical and Chemical Molecular Interactions","score":0.9876000285148621,"subfield":{"id":"https://openalex.org/subfields/1606","display_name":"Physical and Theoretical Chemistry"},"field":{"id":"https://openalex.org/fields/16","display_name":"Chemistry"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10211","display_name":"Computational Drug Discovery Methods","score":0.9750000238418579,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10044","display_name":"Protein Structure and Dynamics","score":0.9614999890327454,"subfield":{"id":"https://openalex.org/subfields/1312","display_name":"Molecular Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/multivariable-calculus","display_name":"Multivariable calculus","score":0.6145144104957581},{"id":"https://openalex.org/keywords/derivative","display_name":"Derivative (finance)","score":0.5851830840110779},{"id":"https://openalex.org/keywords/chain-rule","display_name":"Chain rule (probability)","score":0.562832236289978},{"id":"https://openalex.org/keywords/partial-derivative","display_name":"Partial derivative","score":0.4954909086227417},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.464573472738266},{"id":"https://openalex.org/keywords/subject","display_name":"Subject (documents)","score":0.4471665322780609},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.4115085005760193},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.3145960569381714},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.22860926389694214},{"id":"https://openalex.org/keywords/engineering","display_name":"Engineering","score":0.20692050457000732},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.16805756092071533},{"id":"https://openalex.org/keywords/library-science","display_name":"Library science","score":0.1535976231098175},{"id":"https://openalex.org/keywords/control-engineering","display_name":"Control engineering","score":0.1293012499809265},{"id":"https://openalex.org/keywords/business","display_name":"Business","score":0.1092301607131958},{"id":"https://openalex.org/keywords/biology","display_name":"Biology","score":0.09295979142189026}],"concepts":[{"id":"https://openalex.org/C117312493","wikidata":"https://www.wikidata.org/wiki/Q2035437","display_name":"Multivariable calculus","level":2,"score":0.6145144104957581},{"id":"https://openalex.org/C111771559","wikidata":"https://www.wikidata.org/wiki/Q66295","display_name":"Derivative (finance)","level":2,"score":0.5851830840110779},{"id":"https://openalex.org/C33825631","wikidata":"https://www.wikidata.org/wiki/Q17004731","display_name":"Chain rule (probability)","level":5,"score":0.562832236289978},{"id":"https://openalex.org/C53846429","wikidata":"https://www.wikidata.org/wiki/Q186475","display_name":"Partial derivative","level":2,"score":0.4954909086227417},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.464573472738266},{"id":"https://openalex.org/C2777855551","wikidata":"https://www.wikidata.org/wiki/Q12310021","display_name":"Subject (documents)","level":2,"score":0.4471665322780609},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4115085005760193},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3145960569381714},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.22860926389694214},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.20692050457000732},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.16805756092071533},{"id":"https://openalex.org/C161191863","wikidata":"https://www.wikidata.org/wiki/Q199655","display_name":"Library science","level":1,"score":0.1535976231098175},{"id":"https://openalex.org/C133731056","wikidata":"https://www.wikidata.org/wiki/Q4917288","display_name":"Control engineering","level":1,"score":0.1293012499809265},{"id":"https://openalex.org/C144133560","wikidata":"https://www.wikidata.org/wiki/Q4830453","display_name":"Business","level":0,"score":0.1092301607131958},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.09295979142189026},{"id":"https://openalex.org/C57830394","wikidata":"https://www.wikidata.org/wiki/Q278079","display_name":"Posterior probability","level":3,"score":0.0},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C49698424","wikidata":"https://www.wikidata.org/wiki/Q1989192","display_name":"Law of total probability","level":4,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C107673813","wikidata":"https://www.wikidata.org/wiki/Q812534","display_name":"Bayesian probability","level":2,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/19m1296732","is_oa":true,"landing_page_url":"https://doi.org/10.1137/19m1296732","pdf_url":"https://epubs.siam.org/doi/pdf/10.1137/19M1296732","source":{"id":"https://openalex.org/S160107561","display_name":"SIAM Review","issn_l":"0036-1445","issn":["0036-1445","1095-7200"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Review","raw_type":"journal-article"},{"id":"pmh:oai:archive.ugent.be:8733232","is_oa":true,"landing_page_url":"https://biblio.ugent.be/publication/8733232","pdf_url":null,"source":{"id":"https://openalex.org/S4306400478","display_name":"Ghent University Academic Bibliography (Ghent University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I32597200","host_organization_name":"Ghent University","host_organization_lineage":["https://openalex.org/I32597200"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"ISSN: 1095-7200","raw_type":"info:eu-repo/semantics/publishedVersion"}],"best_oa_location":{"id":"doi:10.1137/19m1296732","is_oa":true,"landing_page_url":"https://doi.org/10.1137/19m1296732","pdf_url":"https://epubs.siam.org/doi/pdf/10.1137/19M1296732","source":{"id":"https://openalex.org/S160107561","display_name":"SIAM Review","issn_l":"0036-1445","issn":["0036-1445","1095-7200"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Review","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3210814943.pdf","grobid_xml":"https://content.openalex.org/works/W3210814943.grobid-xml"},"referenced_works_count":6,"referenced_works":["https://openalex.org/W1581785364","https://openalex.org/W1970408998","https://openalex.org/W2128747615","https://openalex.org/W2139918929","https://openalex.org/W2796669185","https://openalex.org/W2914640698"],"related_works":["https://openalex.org/W1572830206","https://openalex.org/W4313750128","https://openalex.org/W1767607552","https://openalex.org/W2372663877","https://openalex.org/W2060653713","https://openalex.org/W3042042853","https://openalex.org/W2782711679","https://openalex.org/W4245977622","https://openalex.org/W2097846334","https://openalex.org/W4252202378"],"abstract_inverted_index":{"The":[0,70],"multivariable":[1],"chain":[2],"rule":[3],"is":[4,11,72,97],"often":[5],"challenging":[6],"to":[7,45,112],"students":[8],"because":[9],"it":[10],"usually":[12],"presented":[13],"with":[14,66],"ambiguities":[15],"and":[16,22,35,58,87],"other":[17,114],"defects":[18],"that":[19],"hamper":[20],"systematic":[21],"reliable":[23],"application.":[24],"A":[25],"very":[26],"simple":[27],"formulation":[28],"combines":[29],"the":[30,67,78,88,113],"derivation":[31],"operators":[32],"for":[33,36,99],"functions":[34],"expressions":[37],"in":[38,61,102],"a":[39,62,84,100,110],"manner":[40],"not":[41,91],"found":[42],"elsewhere":[43],"due":[44],"common":[46],"confusion":[47],"between":[48],"them.":[49],"Some":[50],"issues":[51],"are":[52,59],"rooted":[53],"more":[54],"deeply":[55],"than":[56],"others":[57],"discussed":[60],"broader":[63],"perspective,":[64],"starting":[65],"function":[68],"concept.":[69],"approach":[71],"illustrated":[73],"using":[74],"various":[75],"applications":[76],"including":[77],"transport":[79],"equation,":[80],"partial":[81,107],"derivatives":[82],"of":[83],"definite":[85],"integral,":[86],"distortionless":[89],"(but":[90],"lossless)":[92],"transmission":[93],"line.":[94],"This":[95],"note":[96],"suitable":[98],"lecture":[101],"any":[103],"first-year":[104],"course":[105,115],"covering":[106],"derivatives,":[108],"as":[109],"complement":[111],"material.":[116]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
