umaC=LDER VL (m1) /ua=-0 The integral form is the form of the differential forms are derived. In contrast with the differential form, we note that the integral from is well defined even when the solution u and or the flur f are discontinuous. We show below, an example of derivation of the different forms of the conservation laws from physical principle. 1.2 Derivation Example 1.2.1Cor p(xt)°b4 ATE OF CHANGE OF MASS FLUX OF FLUID IASS INSIDEΩ THT OUGH aQ V·(pu) t v·(p)a=0 heIt all n,h wacc wtda Th cFtha differential forIt thaichatvat gc lar To derive the differential form of the conservation law, we hat P(a, t)and v(a, t) are differentiable functionó✽ô❦ó✽ôâõ ö✑÷➄ø✭ù➢ú❽û⑦ü✉ý✫þ✜ÿ❽û✁￾ ✂☎✄✝✆✟✞✡✠☞☛ ✌✎✍✑✏✓✒✕✔✗✖✙✘✛✚✢✜✤✣✙✥✓✦★✧✤✖✙✍✪✩✫✜✬✔✭✏✯✮✱✰✳✲✴✓✵✷✶✸✴✓✹✻✺✡✼✯✽✾ ✿❁❀❃❂❅❄☎❆❄☎❇✤❈ ❄❊❉✷❋●❆❊❍ ❄✴❏■▲❑✝▼✳◆P❖ ❑ ❑ ❇ ✿❀ ❆ ❑✝▼✳◆✳◗ ✲❉✷❋●❆✹❍ ◗ ❉✷❋●❆✵ ❍ ✺ ❘❚❙❱❯❳❲✟❨✓❩❬❯❬❭✑❪✸❫✁❴✝❵❜❛✁❪❞❝❡❲✗❢❣❩●❙❱❯❤❝✤❛✁❢❞❩✐❭✝❯✛❨☎❯✛❪✸❫✁❴❁❵❜❛✁❪❞❝❥❛✕❵❣❩●❙❱❯✤❦❧❛✁❨❱❢✛❯✛❪❞♠★❫✬❩♥❲♦❛✬❨♣❴q❫✁r☞❵❧❪✸❛✁❝srt❙❁❲♦❦❧❙ ❩●❙❱❯✈✉✁❲✇①❯✛❪✸❯✛❨②❩③❲♦❫✁❴④❵❜❛✬❪❞❝⑤❢⑥❫✬❪✸❯✈✉✑❯✛❪❞❲✟♠★❯❧✉✑⑦⑨⑧❜❨⑩❦❧❛✁❨✓❩♥❪✸❫✁❢❞❩✡r✐❲✟❩●❙✫❩●❙❱❯✈✉✁❲✇①❯✛❪✸❯✛❨②❩③❲♦❫✁❴★❵❜❛✁❪❞❝❤❶tr❷❯①❨☎❛✬❩❬❯ ❩●❙❱❫✁❩✢❩●❙❱❯✫❲✟❨②❩❸❯③❭✑❪✸❫✬❴❁❵❧❪❹❛✁❝❥❲✗❢❳r❷❯❜❴✟❴❷✉✪❯♦❺❷❨☎❯❧✉❻❯✛♠★❯✛❨❃rt❙❱❯✛❨❃❩✟❙②❯❳❢✛❛✁❴❽❼✙❩③❲♦❛✁❨ ❆ ❫✬❨☎✉❞❾❚❛✁❪✤❩●❙❱❯✢❿➀❼✑➁ ❉ ❫✬❪✸❯✫✉✁❲✗❢✛❦❧❛✁❨✓❩♥❲✟❨✓❼②❛✁❼❁❢✛⑦➃➂✯❯⑤❢✸❙❱❛✬r➅➄❧❯✛❴q❛✁r✷❶➆❫✁❨➇❯❹➁✝❫✬❝➆➈❚❴q❯✫❛➉❵❳✉✑❯✛❪❞❲✟♠★❫✬❩♥❲♦❛✬❨➊❛➉❵⑤❩✟❙②❯✫✉✁❲✇①❯✛❪✸❯✛❨②❩ ❵❜❛✁❪❞❝❤❢⑤❛➉❵➋❩●❙❱❯❳❦❧❛✁❨②❢❜❯✛❪❞♠✁❫✁❩③❲♦❛✁❨➌❴✭❫✁r✷❢❷❵❧❪✸❛✬❝✳➈②❙❁➍★❢❞❲♦❦❧❫✁❴②➈☎❪❞❲✟❨☎❦✛❲➈☎❴✭❯❞❢✛⑦ ➎✐➏♥➐ ➑➓➒→➔✓➣❸↔✷↕t➙❱➣✕➛❷➜➞➝⑩➟✢↕✐➠➢➡⑥➤➉➒ ó✽ô✗➥❇ô❦ó ➦❯ÿ✽÷t➧✭ùrû➩➨✉ü✉ø④➫❦ÿ✽÷❴ÿ☎➭✢➯ü❱➧➩➧ ✂☎✄✝✆✟✞✡✠➌➲ ✌✎✍✑✏✓✒✕✔✗✖✙✘✛✚➆✜➵➳✑✍✑➸✭➺❱✩✤✘⑤✮➻✘✛✏②➼✛➸q✍✝✒✕✘➩✖➾➽❁➚✯✒✕➺②✚✕➪♦✜✪➼❜✘ ❄✮➻➼✛✍✑✏✝➶❹✜✑✔q✏②✔q✏❱➹✤➘✓➺❱✔✭✖✯✍✑➪✷✖❱✘✛✏②✒✸✔➴➶➉➚⑩➷ ❋♦➬ ✶ ❇✕❍ ✜✬✏✓✖➱➮❁✏❱✍✁✃✢✏⑩➳✑✘④➸q✍✙➼❜✔q➶➉➚➵❐ ❋♥➬ ✶ ❇✕❍ ✾➆❒✐❮✢❰PÏ✈ÐÑ✌➀Ò⑥❒①Ó✈ÔÕ❰➻Ï✈Ð ✰ Ö×❒✈Ø❱Ø✯Ð✷Ù✡Ú➆ÛÜÏ✈ÐÝÐ✻ÙtÚ➆✽➉Þ Ö×❒⑥Ø❱Ø⑩✽➉Ó✈Ø❁✽➉Þ✈❰P✮ ❮✢Ò⑥✾➆Ï✈Ú⑥ÔÕÒ ❄✮ ❄ ❄✓❇ ✿❁❀ ➷ ❑✝▼ ◆ ◗ ✿❱ß④❀ ➷✝❐✯à❜á ❑✝â ◆ ◗ ✿❀❳ã à ❋ ➷✝❐ ❍ ❑❁▼ ✂☎✄✝✆✟✞✡✠☞ä ✿❀Ñå❜❄➷❄☎❇❤❈ ã à ❋➷✝❐ ❍❬æ ❑❁▼✳◆➓❖ ç✍✪➸✭✖②✒➀➪●✍✑✚➆✜✬➸✭➸t✮✈è❱✒✸✍✤✃✎✘❣➼④✜✬✏⑩✃✢✚✸✔q➶✸✘ ❄➷❄☎❇❤❈ ã à ❋ ➷✝❐ ❍ ◆P❖ ❮ç✔✗✒✢✔✗✒➀➶ç✘❣é➫●ê✹ù➧û⑦ù➢÷➄ø④➫æü✉ý➀➭ÿ⑤û✁￾ ✍✑➪✡➶ç✘❣➼❜✍✪✏②✒✕✘④✚✸➳✁✜✬➶✸✔✭✍✑✏➱➸✗✜★✃➋ë ❘✓❛➌✉✑❯✛❪❞❲✟♠✁❯➱❩●❙❱❯⑩✉✁❲✇⑥❯❜❪✸❯✛❨②❩③❲♦❫✁❴✓❵❜❛✁❪❞❝ì❛➉❵✤❩●❙❱❯⑩❦❧❛✁❨❱❢✛❯✛❪❞♠★❫✬❩♥❲♦❛✬❨➊❴✭❫✁r✷❶⑥r❷❯❳❙❱❫✬♠★❯✯❫★❢❧❢❞❼✙❝✫❯❹✉×❩✟❙②❫✁❩ ➷ ❋♦➬ ✶ ❇✕❍ ❫✁❨❚✉ ❐ ❋♦➬ ✶ ❇✕❍ ❫✬❪✸❯❤✉✬❲✇①❯✛❪✸❯❜❨✓❩♥❲♦❫✪➄❜❴✭❯✎❵❧❼✙❨❚❦❜❩③❲♦❛✁❨②❢✛⑦ í