17.1-bo'lim : jingalak va farqlanish
2. \(\displaystyle \vec F = \left( {3x + 2{z^2}} \right)\,\vec i + \frac{{{x^3}{y^2}}}{z}\vec j - \left( {z - 7x} \right)\vec k\) uchun \({\mathop{\rm div}\nolimits} \vec F\) va \({\mathop{\rm curl}\nolimits} \vec F\) ni hisoblang.
Yechim
Keling, avval farqni hisoblab chiqamiz va formulani ko'rib chiqishdan boshqa qiladigan ko'p narsa yo'q.
\[{\mathop{\rm div}\nolimits} \vec F = \nabla \centerdot \vec F = \frac{\partial }{{\partial x}}\left( {3x + 2{z^2}} \right) + \frac{\partial }{{\partial y}}\left( {\frac{{{x^3}{y^2}}}{z}} \right) + \frac{\partial }{{\partial z}}\left( {7x - z} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{2 + \frac{{2{x^3}y}}{z}}}\]Ehtiyot bo'ling, har qanday vektor komponentlari (3rd bu holda komponent!). Shoshilish va ularni sog'inish oson.
2-qadam
Jingalak biroz ko'proq ishlaydi, lekin baribir faqat formula ishlaydi, shuning uchun bu erda jingalak.
\[\begin{align*}{\mathop{\rm curl}\nolimits} \vec F& = \nabla \times \vec F = \left| {\begin{array}{*{20}{c}}{\vec i}&{\vec j}&{\vec k}\\{\displaystyle \frac{\partial }{{\partial x}}}&{\displaystyle \frac{\partial }{{\partial y}}}&{\displaystyle \frac{\partial }{{\partial z}}}\\{3x + 2{z^2}}&{\displaystyle \frac{{{x^3}{y^2}}}{z}}&{7x - z}\end{array}} \right|\\ & = \frac{\partial }{{\partial y}}\left( {7x - z} \right)\vec i + \frac{\partial }{{\partial z}}\left( {3x + 2{z^2}} \right)\vec j + \frac{\partial }{{\partial x}}\left( {\frac{{{x^3}{y^2}}}{z}} \right)\vec k\\ & \hspace{0.25in}\hspace{0.25in}\hspace{0.25in}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - \frac{\partial }{{\partial y}}\left( {3x + 2{z^2}} \right)\vec k - \frac{\partial }{{\partial x}}\left( {7x - z} \right)\vec j - \frac{\partial }{{\partial z}}\left( {\frac{{{x^3}{y^2}}}{z}} \right)\vec i\\ & = 4z\vec j + \frac{{3{x^2}{y^2}}}{z}\vec k - 7\vec j + \frac{{{x^3}{y^2}}}{{{z^2}}}\vec i\\ & = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{{x^3}{y^2}}}{{{z^2}}}\vec i + \left( {4z - 7} \right)\vec j + \frac{{3{x^2}{y^2}}}{z}\vec k}}\end{align*}\]Shunga qaramay, 3-dagi minus belgisini unutmangrd komponent.