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16.1-bo'lim : Vektor maydonlari

2. \(\vec F\left( {x,y} \right) = \left( {y - 1} \right)\,\vec i + \left( {x + y} \right)\vec j\) uchun vektor maydonini chizing.

Yechim

Eslatib o'tamiz, vektor maydonining grafigi shunchaki nuqtalar to'plami uchun ma'lum nuqtalardagi vektorlarni chizadi. Bu vektor maydonlarini chizishni oddiy va qiyin qiladi. Vektorlarni hisoblash va ularning eskizini chizish oson, lekin qaysi nuqtalarni tanlash va qancha nuqtani tanlashni bilish qiyin, shuning uchun biz yaxshi eskiz olamiz.

Shunday qilib, keling, ba'zi vektorlarni aniq nuqtalarda hisoblashdan boshlaylik.

\[\begin{align*}\vec F\left( {0, - 1} \right) & = - 2\,\vec i - \vec j & \hspace{0.5in}\vec F\left( {1,1} \right) & = 2\vec j & \hspace{0.5in}\vec F\left( { - 1,0} \right) & = - \,\vec i - \vec j\\ \vec F\left( { - 2, - 1} \right) & = - 2\,\vec i - 3\vec j& \hspace{0.5in}\vec F\left( {1, - 1} \right) & = - 2\,\vec i& \hspace{0.25in}\vec F\left( {2,2} \right) & = \,\vec i + 4\vec j\\ \vec F\left( { - 2,1} \right) & = - \vec j\end{align*}\]
2-qadam

Endi biz ushbu vektorlarning har birini ularni yaratgan nuqtada "eskiz" qilishimiz kerak. Masalan, \(\left( {0, - 1} \right)\) nuqtasida \( - 2\vec i - \vec j\) vektorining eskizini chizamiz.

Mana bu vektorlarning eskizi.

A sketch of the seven vectors computed in the previous step, each drawn with its tail at the point that generated it, on a set of axes running from \(-2\) to 2 in both directions. The vector at \(\left( {2,2} \right)\) is long and points up and slightly right, the one at \(\left( {1,1} \right)\) points straight up, the one at \(\left( { - 2,1} \right)\) is very short and points straight down, the ones at \(\left( { - 1,0} \right)\) and \(\left( { - 2, - 1} \right)\) point down and to the left, the one at \(\left( {0, - 1} \right)\) points left and slightly down, and the one at \(\left( {1, - 1} \right)\) points straight left. The vectors are sketched in the correct direction but not to scale.

Yuqoridagi eskizda biz ushbu vektorlarning har birini masshtablash uchun chizmadik. Boshqacha qilib aytganda, biz vektorni "to'g'ri" kattalik bilan chizish o'rniga, ko'rsatilgan vektor bilan bir xil yo'nalishdagi vektorlarni chizdik. Buning sababi eskizni ko'rishni biroz osonroq saqlashdir. Agar biz barcha vektorlarni masshtablash uchun chizgan bo'lsak, biz shunchaki bir-biriga o'xshash o'qlarni ko'ramiz, bunda nima bo'layotganini ko'rish qiyin bo'ladi.

Shuni ham yodda tutingki, biz chizgan bir nechta vektorlar bilan har qanday tasodifiy nuqtada nima sodir bo'layotganini vektor maydonidagi har qanday tendentsiyalarni haqiqiy his qilish qiyin.

3-qadam

Quyida yana ko'plab vektorlar chizilgan vektor maydonining yaxshiroq eskizi keltirilgan. Biz bu eskizni kompyuterga oldingi bosqichda bo'lgani kabi ulardan hech birini tanlamasdan, o'z-o'zidan bir nechta nuqtalarni chizish imkonini berish orqali oldik.

Umuman olganda, vektor maydonlari shunday chiziladi. Ushbu vektor sonini qo'lda hisoblash shunchalik ko'p vaqt talab qiladiki, bunga arzimaydi. Biroq, kompyuterlar barcha hisob-kitoblarni juda tez bajarishi mumkin va shuning uchun biz ularga eskizni bajarishga ruxsat beramiz.

A computer generated sketch of the vector field \(\vec F\left( {x,y} \right) = \left( {y - 1} \right)\,\vec i + \left( {x + y} \right)\vec j\) with a great many vectors drawn on a grid running from \(-2\) to 2 in both directions. The overall pattern swirls counterclockwise about the point \(\left( { - 1,1} \right)\), where both components are zero and the vectors shrink away to almost nothing. Vectors in the upper right point up and to the right and are the longest in the sketch, those along the bottom point to the left, those in the lower left point down and to the left, and those in the upper left are short and point roughly to the left.