A general partial derivative can be approximated as bmm=am=∑∑bb 户kmb7n()L(y)0 Therefore, defining the weights L i (a)Lk(y) The method of undetermined coefficient extends to multiple dimensions in a straightfor ward manner if we consider multidimensional Taylor series expan assuming a uniform△zand△g △x)22 U +k△ 28 +G△rk△y) azay I(a) A finite difference approximation of the form can be obtained by inserting the Taylor series expansions for ujk and determining Oik coefficients by the coefficients of the different derivative terms ct differe The finite difference approximations considered here are called divided diffe ences approximations. More sophisticated, and accurate, difference approxima tions are also possible. These approximations are called compact approxima ∑v d a ll knowi at ion to the first deriva erify, using Taylor expansi h that this if fourth order accurate for a sufficiently smooth function, 1. e. p=4➽➚➾✷➪❦➶✌➪✳➹✩➘✪➴↕➷✕➘✷➹❝➬❷➮❽➘✷➴→➱✜➪✳➹❷➮❽✃✑➘✪➬❖➮❽✃✷➪➏❐✳➘✷➶P❒✎➪➋➘✪➷✕➷✌➹❖❮✑❰✜➮❽Ï✒➘✭➬❖➪➇➱➁➘✼Ð Ñ↕Ò①Ó→Ô✺Õ Ñ✎ÖÒ Ñ✎× ÔÙØ Ñ✎Ò①Ó→Ô♠ÚÕ Ñ✎ÖÒ Ñ✎ÖÔÜÛ ÝÞ ß❖à➍á✚â Þã äà♠á❅å Ñ↕Ò①Ó→Ô Ñ✎ÖÒ Ñ✚× ÔPæß✷çÖ↕è æä ç×✌è↕Õß äPé ê①ë➪❦➹❖➪✳ì▼❮✷➹❷➪✷í✜➱✜➪✘î✚➶✌➮✾➶✕➾❼➬ë➪➄ïð➪✳➮❽➾ë➬❷Ð ñÒ①Ô ß ä Û ÑÒ①Ó→Ô Ñ✎ÖÒ Ñ✎× ÔPæß çÖ↕è æä ç ×✌è✭òò ò ò✾ó✾ô❦õ✘ö ÷✩õ❝ø ïð➪ ë➘✑✃✷➪ ÑÒðÓ❑Ô ÕÚ Ñ✎ÖÒ Ñ✚× Ô ò ò ò ò✾ó✾ô❦õ✘ö ÷✩õ❝ø➍ù ÝÞ ß❖à♠á✎â Þã äà♠á↕å ñÒ①Ô ß ä Õ ß ä ê①ë➪PÏ❼➪✘➬ë❮✺➱✙❮✷ì❯ú✌➶✕➱✜➪✳➬❖➪❦➹❖Ï❼➮❽➶✌➪❦➱✙❐✘❮✺➪✘û◆❐✳➮✾➪❦➶✼➬✒➪✳❰⑩➬❷➪✳➶✕➱✕Ð➈➬❷❮ÙÏ➈ú✌➴✾➬❖➮❽➷✌➴❽➪✱➱✌➮✾Ï❼➪✳➶✚Ð❝➮❽❮✷➶✕Ð❼➮❽➶✠➘ Ð❝➬❖➹✩➘✪➮❽➾ë➬❝ì▼❮✼➹❖ï①➘✪➹✩➱ÜÏ✒➘✪➶✌➶✕➪✳➹✒➮✾ì➏ïð➪③❐✘❮✼➶✕Ð❖➮❽➱✜➪❦➹✒Ï➈ú✌➴✾➬❖➮✫➱✜➮❽Ï❼➪✳➶✕Ð❖➮✾❮✼➶✕➘✪➴ ê➘✑ü✺➴❽❮✷➹✒Ð❝➪❦➹❖➮❽➪❦Ð❼➪✳❰✜➷✕➘✪➶✜ý Ð❖➮✾❮✼➶✕Ð✳þ ê①ëú✕Ð❦í✌➘✷Ð❷Ð❝ú✕Ï➉➮❽➶✌➾✒➘➉ú✌➶✌➮✾ì▼❮✷➹❷Ï ÿ❉Ö③➘✪➶✚➱Pÿ❉×↕í✌ïð➪ ë➘✑✃✷➪ Õ ß ä Û Õ ó✾ô➇õ✄ö ÷✩õ❖ø ✁✄✂ ÿ❉Ö Ñ✎Õ Ñ✎Ö ò ò ò ò ó✾ô õ ö ÷õ ø ✁✆☎ ÿ➈× Ñ✎Õ Ñ✎× ò ò ò ò✾ó✾ô õ ö ÷õ ø ✁ ç✂ÿ❉Ö✎è✞✝ ✟ Ñ✠✝✘Õ Ñ✎Ö✝ ò ò ò ò✾ó✾ô õ ö ÷õ ø ✁ ç✂ÿ➈Ö ☎ÿ❉×✌è Ñ✡✝✳Õ Ñ✚Ö✎Ñ✎× ò ò ò ò ó✾ô õ ö ÷õ ø ✁ ç ☎ÿ❉×✌è✞✝ ✟ Ñ✠✝✳Õ Ñ✎× ✝ ò ò ò ò ó✾ô õ ö ÷õ ø ✁☞☛✌☛✍☛ é ➽ î✕➶✌➮✾➬❖➪➋➱✌➮✏✎↕➪✳➹❷➪✳➶✕❐✳➪➄➘✪➷✌➷✌➹❷❮✑❰✜➮✾Ï✒➘✪➬❖➮❽❮✷➶➁❮✷ì❋➬ë➪➏ì▼❮✷➹❷Ï ÑÒðÓ❑Ô Õ Ñ✎ÖÒ Ñ✚× Ô ò ò ò ò✾ó✾ô❦õ✘ö ÷✩õ❝ø Ø ÝÞ ß❖à♠á✎â Þã äà♠á↕å ñÒ①Ô ß ä Õ ß ä ❐✳➘✷➶➏❒✚➪r❮✼❒✜➬❷➘✷➮✾➶✕➪❦➱➏❒✺ü❯➮❽➶✕Ð❖➪✳➹❖➬❖➮❽➶✌➾①➬ë➪ ê➘✑ü✺➴❽❮✷➹→Ð❖➪✳➹❷➮✾➪➇Ð❅➪✘❰✜➷✕➘✷➶✕Ð❝➮❽❮✷➶✚Ð↕ì▼❮✷➹→Õß ä ➘✪➶✕➱➏➱✜➪✳➬❖➪✳➹❷Ï❼➮✾➶✕➮✾➶✌➾ ➬ë➪①ñÒ①Ô ß ä ❐✘❮✺➪✘û◆❐✘➮❽➪✳➶⑩➬✩Ð♠❒⑩ü➈➪✌✑⑩ú✕➘✭➬❷➮✾➶✕➾❯➬ë➪❨❐✘❮✺➪✘û◆❐✳➮✾➪❦➶✼➬✩Ð❥❮✪ì✕➬ë➪❨➱✜➮✒✎✎➪❦➹❖➪❦➶⑩➬➍➱✌➪✳➹❷➮✾✃✭➘✭➬❷➮✾✃✼➪◗➬❷➪✳➹❷Ï✒Ð✳þ ✓✕✔✗✖✙✘✛✚ ✜✢✔✤✣✕✥✧✦✡★✤✖✪✩✗✫✭✬✮✘✰✯✱✘✰✲✳★✱✘✴✦✱✥✵✥✶✯✱✔✰✷✸✫✹✣✺✦✗✖✻✫✞✔✤✲✵✼ ê①ë➪✒î✚➶✌➮➬❷➪➁➱✜➮✒✎✎➪❦➹❖➪❦➶✕❐✘➪t➘✪➷✌➷✌➹❷❮✑❰✜➮✾Ï✒➘✪➬❖➮❽❮✷➶✕Ð➏❐✘❮✼➶✕Ð❖➮❽➱✜➪❦➹❖➪➇➱ ë➪✳➹❷➪◆➘✪➹❷➪◆❐✳➘✷➴✾➴❽➪❦➱Ù➱✜➮❽✃✺➮❽➱✜➪➇➱Ü➱✜➮✒✎✎➪❦➹❝ý ➪✳➶✚❐✘➪❦Ð①➘✷➷✌➷✌➹❷❮✑❰✺➮❽Ï✒➘✭➬❷➮✾❮✼➶✕Ð✳þ✵✽✯❮✷➹❷➪➝Ð❖❮✷➷ë➮❽Ð❝➬❖➮✫❐✳➘✪➬❖➪➇➱❅í✌➘✪➶✕➱t➘✷❐❦❐✘ú✌➹✩➘✭➬❷➪✷í✺➱✜➮✏✎↕➪✳➹❷➪✳➶✚❐✘➪➏➘✪➷✌➷✌➹❷❮✑❰✜➮✾Ï✒➘✪ý ➬❖➮❽❮✷➶✚Ð➋➘✪➹❷➪◆➘✪➴✫Ð❝❮✯➷✎❮✼Ð❷Ð❖➮✾❒✌➴❽➪✷þ ê①ë➪➇Ð❝➪◆➘✷➷✌➷✌➹❷❮✑❰✺➮❽Ï✒➘✭➬❷➮✾❮✼➶✕Ð➄➘✪➹❷➪◆❐✳➘✷➴✾➴❽➪❦➱Ù❐✘❮✼Ï❼➷✕➘✷❐✘➬➈➘✪➷✌➷✌➹❷❮✑❰✜➮✾Ï✒➘✪ý ➬❖➮❽❮✷➶✚Ð✮➘✪➶✕➱t➬❷➘✿✾✷➪➝➬ë➪➏ì▼❮✼➹❖Ï ÝÞ❀ ß❖à♠á✎â ❀ ❁Ò Õ ❁ÖÒ ò ò ò ò ô à ô✌❂ Ø ÝÞ ß❖à♠á✎â ñÒ ß Õ ß❄❃ ➘✪➶✚➱t➘➉➷✕➘✷➹❝➬❷➮❽❐✳ú✌➴❽➘✷➹✮ï✁➪❦➴✾➴✗✾✺➶✌❮✭ï✮➶✱❐✘❮✼Ï❼➷✕➘✷❐✘➬✮➘✷➷✌➷✌➹❷❮✑❰✜➮✾Ï✒➘✭➬❷➮✾❮✼➶◆➬❷❮❉➬ë➪➏î✕➹✩Ð✥➬❨➱✜➪❦➹❖➮❽✃✭➘✭➬❷➮✾✃✼➪➎➮✫Ð ❅ ❆ çÕ✤❇ß Ó❉❈ ✁❋❊Õ●❇ß ✁ Õ✤❇ß✘á ❈ è Û Õ ß Ó❉❈■❍ Õ ß✘á ❈ ✟ÿ❉Ö é ❏➬❑➮✫Ð→➪➇➘✷Ð❖ü❨➬❖❮❯✃✷➪✳➹❷➮✾ì▼ü✷í➇ú✕Ð❝➮❽➶✌➾ ê➘✑ü✺➴✾❮✼➹❅➪✘❰✜➷✕➘✷➶✕Ð❝➮❽❮✷➶✚Ð❅❮✷➹→❮✷➬ë➪❦➹❖ï✮➮✫Ð❖➪✷í❦➬ë➘✪➬❑➬ë➮❽Ð❋➘✪➷✌➷✌➹❷❮✑❰✜➮✾Ï✒➘✪➬❖➮❽❮✷➶ ➮✾ì❋ì▼❮✼ú✌➹❝➬ë ❮✼➹❷➱✌➪✳➹❨➘✷❐❦❐✘ú✌➹✩➘✭➬❷➪❯ì▼❮✼➹❨➘❼Ð❝ú✜û◆❐✳➮✾➪❦➶✼➬❷➴✾üPÐ❝Ï❼❮✺❮✪➬ë ì▼ú✕➶✕❐✄➬❷➮✾❮✼➶→í✌➮✉þ ➪✼þ✗❑ Û ❊ þ ▲