12.10-bo'lim : Egrilik
2. \(\vec r\left( t \right) = \left\langle {4t, - {t^2},2{t^3}} \right\rangle \) ning egri chizig‘ini toping.
Yechim
Egrilikni hisoblash uchun bizda ikkita formuladan foydalanishimiz mumkin. Ulardan biri birlik tangens vektorining hosilasini olishimizni talab qiladi, ikkinchisi esa o'zaro ko'paytmani talab qiladi.
Ikkalasi ham bir xil natija beradi. Haqiqiy savol - qaysi biri foydalanish osonroq bo'ladi. O'zaro mahsulotlarni hisoblash qiyin bo'lishi mumkin, ammo keyin birlik tangens vektorlarining hosilasini olish juda tartibsiz bo'lishi mumkin. Shunday qilib, asosan, biz foydalanadigan narsa, ehtimol, foydalanish uchun eng oson bo'ladi.
Bunday holda, birlik tangens vektori hosilasini olish yoqimsiz bo'lishi mumkin bo'lgan ko'plab bo'limlarni o'z ichiga oladi. Shunday qilib, keling, bu safar o'zaro mahsulot formulasiga o'taylik.
Bizga vektor funktsiyasining birinchi va ikkinchi hosilalari kerak bo'ladi. Here are those.
\[\vec r'\left( t \right) = \left\langle {4, - 2t,6{t^2}} \right\rangle \hspace{0.5in}\vec r''\left( t \right) = \left\langle {0, - 2,12t} \right\rangle \]2-qadam
Keyinchalik, bizga ushbu ikki hosilaning o'zaro mahsuloti kerak. Mana o'sha ish.
\[\vec r'\left( t \right) \times \vec r''\left( t \right) = \left| {\begin{array}{*{20}{c}}{\vec i}&{\vec j}&{\vec k}\\4&{ - 2t}&{6{t^2}}\\0&{ - 2}&{12t}\end{array}} \right| = - 24{t^2}\,\vec i - 8\vec k - 48t\,\vec j + 12{t^2}\,\vec i = - 12{t^2}\,\vec i - 48t\,\vec j - 8\vec k\]3-qadam
Endi bizga bir nechta kattalik kerak.
\[\left\| {\vec r'\left( t \right) \times \vec r''\left( t \right)} \right\| = \sqrt {144{t^4} + 2304{t^2} + 64} \hspace{0.25in}\hspace{0.25in}\left\| {\vec r'\left( t \right)} \right\| = \sqrt {16 + 4{t^2} + 36{t^4}} \]Keyin egrilik,
\[\kappa = \frac{{\left\| {\vec r'\left( t \right) \times \vec r''\left( t \right)} \right\|}}{{{{\left\| {\vec r'\left( t \right)} \right\|}^3}}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{\sqrt {144{t^4} + 2304{t^2} + 64} }}{{{{\left( {16 + 4{t^2} + 36{t^4}} \right)}^{\frac{3}{2}}}}}}}\]Bu erda juda chalkash formula, lekin bu ko'pincha shunday bo'ladi.