In general we will start with k= 0, and increase k until a sufficient number of independent equations is generated so that all the coefficients are uniquely Soly to recover the same second order central difference approximation to a second derivative previously derived. Note 3 Multidimensional finite difference formulas The Lagrange interpolation and undetermined coefficients approach we have seen can be easily extended to multiple dimensions Lagrange interpolants can be constructed in multiple dimensions by combining one dimensional Lagrange polynomials. Given a lattice of (L+r+1)x(d+u+1) points truct the fol polynomials for a ty point j, k 1)…( LK(y) (y-y-a)…(y-孙k-1)(y-yk+1)…(y-y) (yk-y-a)…(yk-yk-1)(yk-yk+1)…(yk-ya) The lagi interpolant is thus obt ained 0(x,y)=∑∑L3(x)L(y) L ()Li(y)÷✜ø♣ù✄ú☛ø➠ú☛û✔ü✡ý✍þ➬ú✾þ❀ÿ✛ý✛ý✁￾✄✂ü✡û✄✂❘þ❀ÿ☎✂☎✆✞✝✠✟☛✡✌☞ ü✡ø✎✍ ÿ✛ø✎✏✕û✔ú✖ü✑￾✕ú✒✝✔✓☎ø✕✂☞ÿ✛ý➯ü✖￾✄✓✘✗✙✏✕ÿ❊ú☛ø✚✂❘ø✕✓✜✛✒✢✟ú✕û ✣✥✤ ÿ✛ø✎✍✧ú✧✦✐ú✕ø✎✍❳ú✕ø✕✂❫ú✩★✘✓✐ü✪✂✘ÿ ✣ ø✌￾❩ÿ✫￾❈ù✧ú✕ø✓ú✕û✔ü✪✂ú✩✍✬￾ ✣ ✂☎✆✐ü✭✂❻ü✡ý✛ý✮✂☎✆✐ú✯✏✣ú✰✗✙✏✕ÿ❊ú✕ø✕✂✰￾ ü✡û✔ú✱✓☎ø✐ÿ✲★✳✓✐ú✕ý✵✴ ✍❳ú✳✂⑥ú☛û✄✛➉ÿ✛ø✓ú✩✍✷✶ ✸✺✹✼✻✾✽✼✿✷❀ ❁✭❂❃❅❄ ✟ ❆ ❇❉❈❂❋❊ ❁✭❂● ✟■❍ ❏ ❇❉❈❂❑❊ ❁✭❂❄ ✟ ❆ ❇❉❈❂ ✂✣ û✔ú✩✏✣✭▲ ú☛û✒✂▼✆Pú◆￾✕ü✭✛❈ú✒￾✕ú✩✏✣ø✎✍ ✣ û❖✍✧ú☛ûP✏✟ú✕ø✕✂☞û✔ü✡ý◗✍✁ÿ❘sú✕û✔ú✕ø✎✏✖ú❩ü✩✦✷✦➠û✣✳❙ÿ☎✛❈ü✭✂☞ÿ ✣ø❚✂✣ ü✱￾✕ú✩✏✣ø✎✍ ✍❳ú✕û✜ÿ▲ ü✭✂☞ÿ▲ ú❯✦➠û✔ú ▲ÿ ✣✓✺￾✜ý✵✴P✍❳ú✕û✜ÿ▲ ú✩✍✷✶ ❱❯❲ ❱❨❳ ❩✞❬❅❭✄❪✯❫ ❴❛❵✚❜❝❭❖❞❢❡❅❞✲❣❚❪✪❤❥✐❦❞✥❬✺❤♠❧✜❜♠♥♠❤❋❞❖❭✄❪✯❡❅❞▼♦✮❪✪♣✼❪✪❤rq✼❪ts❝❬✺♣✑❣❚❵✕❜✉❧✜✐ ✈①✇✿③②❋④✪⑤✷⑥✩④✪⑦✕⑤✷✿P⑧✾⑦⑩⑨❖✿❦⑥❖❶✎✹✷✻✫④✭⑨❷⑧✾✹✼⑦❸④✪⑦✕❹✙❺✌⑦✕❹✌✿✘⑨❖✿❦⑥❖❻❼⑧❽⑦✌✿❦❹❸❾✳✹✺✿✘❿◆❾✘⑧❽✿✳⑦⑩⑨❷➀➁④✪❶✌❶✌⑥❷✹✼④✼❾✇✠➂✿ ✇④✑✽✼✿ ➀❖✿✳✿✳⑦✱❾✳④✷⑦t➃✎✿➄✿❦④✼➀❝⑧❽✻❽➅◆✿✘➆✺⑨❖✿❦⑦✕❹✜✿➇❹➁⑨❷✹❼❻➈❺✌✻✾⑨❖⑧❽❶✌✻❽✿➈❹✜⑧❽❻➉✿❦⑦✕➀❖⑧✾✹✼⑦✕➀✳➊ ②❑④✷⑤✷⑥✩④✪⑦✌⑤✼✿❨⑧❽⑦✼⑨❷✿✳⑥❷❶✚✹✼✻❽④✷⑦⑩⑨❷➀✮❾✳④✷⑦➁➃✎✿➋❾✘✹✼⑦✕➀✥⑨❷⑥❖❺✚❾✄⑨❖✿➇❹➁⑧❽⑦P❻➈❺✌✻✾⑨❖⑧❽❶✌✻❽✿➋❹✜⑧✾❻❼✿❦⑦✕➀❝⑧❽✹✷⑦✚➀①➃⑩➅t❾✘✹✷❻❉➃✌⑧❽⑦✌⑧✾⑦✕⑤ ✹✷⑦✕✿①❹✌⑧✾❻❼✿✳⑦✚➀❝⑧❽✹✷⑦✕④✷✻✌②❑④✷⑤✷⑥✩④✪⑦✌⑤✼✿◗❶✎✹✷✻❽➅⑩⑦✕✹✷❻❼⑧❽④✷✻❽➀❦➊➍➌➎⑧✾✽✼✿✳⑦❼④➏✻❽④✪⑨❝⑨❖⑧✫❾✘✿①✹✷➐♠➑✰➒✳➓◆➔→➓ ❆➇➣↕↔ ➑✰➙❥➓➁➛①➓ ❆➇➣ ❶✎✹✷⑧❽⑦✼⑨✩➀ ➂✿①❾❦④✪⑦✒❾✘✹✼⑦✕➀❝⑨❖⑥❷❺✕❾✄⑨➍⑨✇✿①➐▼✹✷✻❽✻✾✹➂⑧❽⑦✌⑤➏✹✷⑦✕✿✘➜❢❹✜⑧✾❻❼✿❦⑦✕➀❝⑧❽✹✷⑦✚④✪✻✌②❑④✷⑤✷⑥✩④✪⑦✌⑤✼✿✁❶✎✹✷✻❽➅✺⑦✌✹✷❻❼⑧✫④✪✻✫➀❥➐▼✹✷⑥◗④➝⑨✥➅✺❶✜➜ ⑧✫❾✳④✪✻❅❶✎✹✷⑧❽⑦⑩⑨✁➞ ❊ ✝ ➟◗➠➡ ➑❈ ➣ ✟ ➑❈ ❍ ❈ ❃✚➢ ➣↕➤❦➤✳➤ ➑❈ ❍ ❈ ➡ ❃❅❄ ➣ ➑❈ ❍ ❈ ➡❖➥ ❄ ➣↕➤❦➤✳➤ ➑❈ ❍ ❈✚➦ ➣ ➑❈ ➡ ❍ ❈ ❃→❄ ➣↕➤❦➤✳➤ ➑❈ ➡ ❍ ❈ ➡ ❃→❄ ➣ ➑❈ ➡ ❍ ❈ ➡❖➥ ❄ ➣❅➤✳➤✳➤ ➑❈ ➡ ❍ ❈➦ ➣ ➟✁➧➨ ➑▼➩ ➣ ✟ ➑▼➩ ❍ ➩ ❃↕➫ ➣↕➤✳➤❦➤ ➑▼➩ ❍ ➩➨ ❃❅❄ ➣ ➑✲➩ ❍ ➩➨➥ ❄ ➣↕➤✳➤❦➤ ➑✲➩ ❍ ➩✷➭ ➣ ➑▼➩➨ ❍ ➩ ❃↕➫ ➣↕➤❦➤✳➤ ➑✲➩➨ ❍ ➩➨ ❃❅❄ ➣ ➑✲➩➨ ❍ ➩➨ ➥ ❄ ➣❅➤✳➤✳➤ ➑▼➩➨ ❍ ➩➭ ➣ ✈①✇✿➄②❑④✷⑤✷⑥✩④✪⑦✌⑤✼✿➝⑧❽⑦⑩⑨❖✿❦⑥❖❶✎✹✷✻✫④✪⑦⑩⑨①⑧❽➀①⑨✇❺✕➀✮✹✷➃✌⑨❷④✪⑧❽⑦✌✿➇❹t④✼➀ ➯➲ ➑❈ ❊ ➩ ➣ ✟ ➦ ➳ ➡❖➵ ❃✎➢ ➭ ➳ ➨➵ ❃↕➫ ➟➠ ➡ ➑❈ ➣ ➟◗➧➨ ➑▼➩ ➣ ➲➡ ➨P➸ ➺✿➈⑦✌✹✷⑨❖✿➋⑨✇④✭⑨➎➃✺➅✱❾✳✹✷⑦✕➀❝⑨❖⑥❷❺✕❾✘⑨❖⑧❽✹✷⑦ ➟➠ ➡ ➑❈ ➣ ➟◗➧➨ ➑▼➩ ➣ ⑨✩④✪➻✷✿➇➀❯④➁✽✑④✷✻✾❺✕✿➋✹✪➐r✹✼⑦✌✿❉④✪⑨➉➑❈ ➡ ❊ ➩➨ ➣ ④✪⑦✚❹ ➼✿❦⑥❖✹❼④✭⑨❯④✷✻✾✻❅✹✷⑨✇✿❦⑥✮❶✚✹✼⑧✾⑦⑩⑨✩➀✮⑧✾⑦t⑨✇✿➄✻❽④✪⑨❝⑨❖⑧✫❾✘✿✼➊ ❳