13.6-bo'lim : Zanjir qoidasi
8. Quyidagi vaziyat uchun \(\displaystyle \frac{{\partial w}}{{\partial t}}\) va \(\displaystyle \frac{{\partial w}}{{\partial u}}\) formulalarini aniqlang.
\[w = w\left( {x,y,z} \right)\hspace{0.5in}x = x\left( t \right),\,\,\,\,y = y\left( {u,v,p} \right),\,\,\,\,z = z\left( {v,p} \right),\,\,\,\,v = v\left( {r,u} \right),\,\,\,\,p = p\left( {t,u} \right)\]Yechim
Ushbu hosilalarning formulasini aniqlash uchun bizga quyidagi daraxt diagrammasi kerak bo'ladi.
Ushbu daraxt diagrammalarining ba'zilari juda chalkash bo'lishi mumkin. Biz har bir lotin uchun amal qilishimiz kerak bo'lgan novdalarni biroz aniqroq qilish uchun bizni qiziqtirgan o'zgaruvchilarni rangladik.
Bundan tashqari, shoxlarning oxirgi "qatori" bir-biriga biroz yaqinlashib qolganligi sababli, biz har bir filial bilan qaysi lotin bog'langanligini ko'rishni osonlashtirish uchun pastki lotin yozuviga o'tdik.
2-qadam
Mana bizdan topishimiz so'raladigan formulalar.
\[\frac{{\partial w}}{{\partial t}} = \frac{{\partial w}}{{\partial x}}\frac{{\partial x}}{{\partial t}} + \frac{{\partial w}}{{\partial y}}\frac{{\partial y}}{{\partial p}}\frac{{\partial p}}{{\partial t}} + \frac{{\partial w}}{{\partial z}}\frac{{\partial z}}{{\partial p}}\frac{{\partial p}}{{\partial t}}\] \[\frac{{\partial w}}{{\partial u}} = \frac{{\partial w}}{{\partial y}}\frac{{\partial y}}{{\partial u}} + \frac{{\partial w}}{{\partial y}}\frac{{\partial y}}{{\partial v}}\frac{{\partial v}}{{\partial u}} + \frac{{\partial w}}{{\partial y}}\frac{{\partial y}}{{\partial p}}\frac{{\partial p}}{{\partial u}} + \frac{{\partial w}}{{\partial z}}\frac{{\partial z}}{{\partial v}}\frac{{\partial v}}{{\partial u}} + \frac{{\partial w}}{{\partial z}}\frac{{\partial z}}{{\partial p}}\frac{{\partial p}}{{\partial u}}\]