资源描述：
《化工应用数学.pdf》由会员上传分享，免费在线阅读，更多相关内容在行业资料-天天文库。
1、本文由yw8p62up0h贡献ppt文档可能在WAP端浏览体验不佳。建议您优先选择TXT，或下载源文件到本机,查看。化工應用數學授課教師：郭修伯PartialDifferentiationandPartialDifferentialEquationsLecture7Chapter8PartialdifferentiationandP.D.E.s–Problemsrequiringthespecificationofmorethanoneindependent,variable.–Example,
2、thechangeoftemperaturedistributionwithinasyst,em:Thedifferentiationprocesscanbeperformedrelativetoanincrement,alchangeinthespacevariablegivingatemperaturegradient,orrateof,temperaturerise.PartialderivativesFigure8.1(contourmapforu)–Ifxisallowedtovary
3、whilstyremainsconstanttheningeneral,uwillalsovaryandthederivateofuw.r.t.xwillbetherateofchange,ofurelativetox,orthegradientinthechosendirection:?u?????x?yuisavectoralongthelineofgreatestslopeandhasamagnitudeeq,ualtothatslope.u=i?u?u+j=gradu?x?yui??u=
4、?x?u???δyduetothechangeiny:??y????u?uwillchangeby??δxduetothechangeinx,andby??x??u???u?δu=??δx+??δy??y???x???δu→duδx→dxδy→dy?u???u?du=??dx+??dy??y???x???the“totaldifferential”ofu?u???u???u??dx1+??dx2+⋯⋯+?du=?Ingeneralform:,??x???x???x?dxn??1??2??n?Im
5、portantfactconcerning“partialderivative”Thesymbol“?g“cannotbecancelledout!?Thetwopartsofthera,tiodefiningapartialderivativecanneverbeseparatedandconsideredal,one.–Markedcontrasttoordinaryderivativeswheredx,dycanbetreate,dseparatelyChangingtheindepend
6、entvariablesz=g(x,y)x=φ(u,v)y=ψ(u,v)?g???g?dz=??dx+??dy??y???x????φ???φ?dx=??du+??dv??u???v???ψ???ψ?dy=?,du+???dv??u???v?φ???g???ψ?ψ???g???φdz=???du+dv?+???du+dv,???y??u?v?????v???x???uw.r.t.u?z??z???x???z???y,?=????+?????u??x???u???y???u????z??g???φ
7、???,g???ψ?=????+?????u??x???u???y???u????z???xn??z??z???x1???z???x2????+????+⋯⋯+?=?,Ingeneralform:??x???u???x???u????x???u????u1?1??1,??2??1??n??1?page1IndependentvariablesnottrulyindependentVapourcompositionisafunctionoftemperature,pressureandliquid
8、,composition:y=f(T,P,x)However,boilingtemperatureisafunctionofpressureandliquidcom,posision:T=g(P,x)Therefore?f???f???f?dy=??dx+??dT+??dP??x???T???P??g???g?dT=??dx+??dP??x???P??f???f???g?????f???f???g??dy=???+?????dx+??,?+?????dP???x???T???x?????P???
