# Chapter 14 Notes.pdf - Partial derivatives functions of...

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Partialderivativesfunctionsof severalvariables14.1supposeDisa set ofntables ofreal numbersIXX2XnArealvaluedfunctionsonDisarulethatassignsaUnionesinglereal numberWfXX2XnttoeachelementDThesetDisthe function's domainTheset ofWvaluestakenonbyf isthefunction'srangewisadependent variableoffandfissaidtobethefunctionofthenindependentvariablesxtoxn Wealsocall thexi'sthefunction's input variablesandcall w thefunction'soutputvariableclosedintervalsCabcontain boundarypointsopenintervalsaibdon'ttouchboundarypointsinteriorpointapointix gotinregionR thathasacenterofpositiveradius that liesentirelyinRboundarypointlxYocontainspointsthatliecompletelyoutsideraswell as inoron Ropenregionconsistsentirelyofinteriorpointsclosedregion containsallitsboundary pointsaregion intheplaneisboundedifit lies insideadiskoffiniteradiusAregionisunbounded ifitlacksboundarythe set ofpointsintheplane wherefxyhasaconstantvaluefxyleiscalledthelevelcurveoffthesetofallPointsXyfxyinspaceforxyin thedomainoffis calledthegraphoffThegraphoffisalsocalledthesurface2fix y
examplesgraph fixy100x2yandplot thelevelcurvesfixyo.flx.us51andjlxy75inthedomainoffin the plane2100x2y2fXy100x24207242100810fx424251rFrsobSoverheadeVysideviewsurface2100x242yExlevelsurfacetheset ofpointsix42inspace whereafunctionof3indvariableshasa constantvalueIX4,276

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