16.1-bo'lim : Vektor maydonlari
4. \(\displaystyle f\left( {x,y,z} \right) = {z^2}{{\bf{e}}^{{x^{\,2}} + 4y}} + \ln \left( {\frac{{xy}}{z}} \right)\) uchun gradient vektor maydonini hisoblang.
Yechim
Haqiqatan ham bu muammoni hal qilish uchun ko'p narsa yo'q. Bu funksiya uchun gradient vektor maydoni.
\[\require{bbox} \bbox[2pt,border:1px solid black]{{\nabla f = \left\langle {2x{z^2}{{\bf{e}}^{{x^{\,2}} + 4y}} + \frac{1}{x},4{z^2}{{\bf{e}}^{{x^{\,2}} + 4y}} + \frac{1}{y},2z{{\bf{e}}^{{x^{\,2}} + 4y}} - \frac{1}{z}} \right\rangle }}\]Bularning har biri uchun xususiy hosilalarni hisoblashni unutmang! Birinchi a’zo funksiyaning \(x\) ga nisbatan hosilasi, ikkinchi a’zosi funksiyaning \(y\) ga nisbatan hosilasi va uchinchi had – funktsiyaning \(z\) ga nisbatan hosilasidir.