{"id":"https://openalex.org/W2485644474","doi":"https://doi.org/10.1117/3.581869.ch5","title":"Vector Calculus Integral Forms","display_name":"Vector Calculus Integral Forms","publication_year":2009,"publication_date":"2009-09-04","ids":{"openalex":"https://openalex.org/W2485644474","doi":"https://doi.org/10.1117/3.581869.ch5","mag":"2485644474"},"language":"en","primary_location":{"id":"doi:10.1117/3.581869.ch5","doi":"https://doi.org/10.1117/3.581869.ch5","is_oa":false,"landing_page_url":"https://doi.org/10.1117/3.581869.ch5","pdf_url":null,"source":{"id":"https://openalex.org/S4306463935","display_name":"SPIE eBooks","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/P4310315543","host_organization_name":"SPIE","host_organization_lineage":["https://openalex.org/P4310315543"],"host_organization_lineage_names":["SPIE"],"type":"ebook platform"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Field Mathematics for Electromagnetics, Photonics, and Materials Science","raw_type":"book-chapter"},"parent_works":[{"id":"https://openalex.org/W4248148290","display_name":"Field Mathematics for Electromagnetics, Photonics, and Materials Science: A Guide for the Scientist and Engineer","relationship":"volume"}],"lineage":["https://openalex.org/W2485644474","https://openalex.org/W4248148290"],"child_works_count":0,"type":"book-chapter","is_traditional":true,"indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.0,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":"5-1","last_page":"5-32"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11683","display_name":"Quantum and Classical Electrodynamics","score":0.5917999744415283,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11683","display_name":"Quantum and Classical Electrodynamics","score":0.5917999744415283,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T13806","display_name":"Mathematical and Computational Methods","score":0.1298999935388565,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10739","display_name":"Electromagnetic Scattering and Analysis","score":0.07850000262260437,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/vector-calculus","display_name":"vector calculus","score":0.9419999718666077},{"id":"https://openalex.org/keywords/integral-theorems","display_name":"integral theorems","score":0.7799999713897705},{"id":"https://openalex.org/keywords/green-s-identity","display_name":"Green's identity","score":0.8399999737739563},{"id":"https://openalex.org/keywords/gauss-s-law","display_name":"Gauss's law","score":0.7889999747276306},{"id":"https://openalex.org/keywords/stokes-theorem","display_name":"Stokes' theorem","score":0.8939999938011169},{"id":"https://openalex.org/keywords/maxwell-s-equations","display_name":"Maxwell's equations","score":0.9039999842643738},{"id":"https://openalex.org/keywords/electromotive-force","display_name":"electromotive force","score":0.6520000100135803},{"id":"https://openalex.org/keywords/magnetomotive-force","display_name":"magnetomotive force","score":0.9869999885559082},{"id":"https://openalex.org/keywords/electric-flux-density","display_name":"electric flux density","score":0.675000011920929},{"id":"https://openalex.org/keywords/magnetic-flux-density","display_name":"magnetic flux density","score":0.972000002861023},{"id":"https://openalex.org/keywords/green-s-function","display_name":"Green's function","score":0.7229999899864197},{"id":"https://openalex.org/keywords/nonhomogeneous-differential-equations","display_name":"nonhomogeneous differential equations","score":0.9160000085830688}],"concepts":[{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.6895455121994019},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4783708155155182},{"id":"https://openalex.org/C5408677","wikidata":"https://www.wikidata.org/wiki/Q200802","display_name":"Vector calculus","level":2,"score":0.410385400056839},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.1972174346446991},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.11572861671447754},{"id":"https://openalex.org/C199343813","wikidata":"https://www.wikidata.org/wiki/Q12128","display_name":"Dentistry","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1117/3.581869.ch5","doi":"https://doi.org/10.1117/3.581869.ch5","is_oa":false,"landing_page_url":"https://doi.org/10.1117/3.581869.ch5","pdf_url":null,"source":{"id":"https://openalex.org/S4306463935","display_name":"SPIE eBooks","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/P4310315543","host_organization_name":"SPIE","host_organization_lineage":["https://openalex.org/P4310315543"],"host_organization_lineage_names":["SPIE"],"type":"ebook platform"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Field Mathematics for Electromagnetics, Photonics, and Materials Science","raw_type":"book-chapter"}],"best_oa_location":null,"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W606278518","https://openalex.org/W2492493396","https://openalex.org/W4388578741","https://openalex.org/W4312422548","https://openalex.org/W4226221627","https://openalex.org/W2149636991","https://openalex.org/W2388845057","https://openalex.org/W2351180711","https://openalex.org/W4311687294","https://openalex.org/W2903810931"],"abstract_inverted_index":{"There":[0],"is":[1,163,202],"an":[2,181],"intimate":[3],"relationship":[4],"between":[5,125],"differential":[6,33,126,189],"and":[7,26,50,54,64,108,115,127,149],"integral":[8,44,73,128],"forms":[9,45,103,144],"in":[10,112,196],"vector":[11,32],"calculus":[12,15],"(and":[13],"tensor":[14],"as":[16,106,145],"well).":[17],"For":[18],"example,":[19],"Maxwell's":[20,58],"curl":[21],"equations":[22,60],"for":[23,61,123,184,199],"time-varying":[24,40],"electric":[25,63],"magnetic":[27,65],"field":[28],"intensities,":[29],"which":[30,42,179],"are":[31,43],"operators,":[34],"convert":[35,70],"to":[36,71,120,139,187,204],"circulations":[37],"of":[38,84,89,98,104,147,167,169,212],"these":[39],"fields,":[41],"that":[46,160,206],"describe":[47,140],"the":[48,62,87,99,121,141,158,176],"electromotive":[49],"magnetomotive":[51],"force":[52],"(volts":[53],"amps),":[55],"respectively.":[56],"Further,":[57],"divergence":[59],"flux":[66],"densities":[67],"(differential":[68],"forms)":[69],"closed-surface":[72],"forms.":[74,129],"These":[75],"conversion":[76],"relationships":[77],"can":[78],"be":[79,137,194],"developed":[80],"from":[81,86],"a":[82,164],"series":[83],"theorems":[85],"mathematics":[88,173],"George":[90],"Green":[91],"(1828)":[92],"called":[93],"Green's":[94,143,170,172,177],"identities.":[95,171],"Other":[96],"mathematicians":[97],"1800s":[100],"contributed":[101],"various":[102],"identities\u2014such":[105],"Gauss'":[107,131],"Stokes'":[109,151,161],"theorems,":[110],"discussed":[111],"Sections":[113],"5.3":[114],"5.4,":[116],"respectively\u2014that":[117],"significantly":[118],"add":[119],"tools":[122],"converting":[124],"Since":[130],"work":[132],"preceded":[133],"Green's,":[134,154],"it":[135,201],"would":[136],"accurate":[138],"relevant":[142],"generalizations":[146],"Gauss';":[148],"since":[150],"theorem":[152,162],"followed":[153],"one":[155,168],"could":[156],"take":[157],"position":[159],"special":[165],"case":[166],"also":[174],"included":[175],"function,":[178],"provides":[180,210],"effective":[182],"method":[183],"determining":[185],"solutions":[186],"inhomogeneous":[188],"equations.":[190],"This":[191],"process":[192],"will":[193],"covered":[195],"Section":[197],"5.5;":[198],"now,":[200],"sufficient":[203],"say":[205],"this":[207,213],"tool":[208],"further":[209],"evidence":[211],"differential-integral":[214],"relationship.":[215]},"counts_by_year":[],"updated_date":"2026-11-15T12:55:33.944570","created_date":"2016-08-23T00:00:00"}
