13.6-bo'lim : Zanjir qoidasi
7. Quyidagi vaziyat uchun \(\displaystyle \frac{{\partial w}}{{\partial t}}\) va \(\displaystyle \frac{{\partial w}}{{\partial v}}\) formulalarini aniqlang.
\[w = w\left( {x,y} \right)\hspace{0.5in}x = x\left( {p,q,s} \right),\,\,\,\,y = y\left( {p,u,v} \right),\,\,\,\,s = s\left( {u,v} \right),\,\,\,\,p = p\left( t \right)\]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.
2-qadam
Mana bizdan topishimiz so'raladigan formulalar.
\[\frac{{\partial w}}{{\partial t}} = \frac{{\partial w}}{{\partial x}}\frac{{\partial x}}{{\partial p}}\frac{{dp}}{{dt}} + \frac{{\partial w}}{{\partial y}}\frac{{\partial y}}{{\partial p}}\frac{{dp}}{{dt}}\hspace{0.5in}\frac{{\partial w}}{{\partial v}} = \frac{{\partial w}}{{\partial x}}\frac{{\partial x}}{{\partial s}}\frac{{\partial s}}{{\partial v}} + \frac{{\partial w}}{{\partial y}}\frac{{\partial y}}{{\partial v}}\]