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17.5-bo'lim: Stoks teoremasi

  1. \( \displaystyle \iint\limits_{S}{{{\mathop{\rm curl}\nolimits} \vec F\centerdot d\vec S}}\) ni baholash uchun Stokes teoremasidan foydalaning, bunda \(\vec F = y\,\vec i - x\,\vec j + y{x^3}\,\vec k\) va \(S\) 4 radiusli sharning \(z \ge 0\) va yuqoriga yo‘naltirilgan qismidir. Yechim
  2. \( \displaystyle \iint\limits_{S}{{{\mathop{\rm curl}\nolimits} \vec F\centerdot d\vec S}}\) ni baholash uchun Stokes teoremasidan foydalaning, bunda \(\vec F = \left( {{z^2} - 1} \right)\,\vec i + \left( {z + x{y^3}} \right)\,\vec j + 6\,\vec k\) va \(S\) \(x = - 2\) oldidagi \(x = 6 - 4{y^2} - 4{z^2}\) ning manfiy \(x\) o'qi yo'nalishiga yo'naltirilgan qismidir. Yechim
  3. \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\) ni baholash uchun Stokes teoremasidan foydalaning, bunda \(\vec F = - yz\,\vec i + \left( {4y + 1} \right)\,\vec j + xy\,\vec k\) va \(C\) \(y = 4\) da 3 radiusli va \(y\) o'qiga perpendikulyar bo'lgan doiradir. Agar siz \(y\) o'qini musbat \(y\) o'qidan manfiy \(y\) o'qigacha pastga qaratsangiz, \(C\) soat yo'nalishi bo'yicha aylanadi. Egri chiziqning eskizi uchun quyidagi rasmga qarang.
    A three dimensional sketch of the curve \(C\). It is a circle of radius 3 that lies in a plane perpendicular to the y-axis and is centered on the y-axis at \(y = 4\). The arrows on the circle show that it has a clockwise rotation when looking at it from the positive y-axis back towards the origin.
    Yechim
  4. \( \displaystyle \int\limits_{C}{{\vec F\centerdot d\vec r}}\)ni baholash uchun Stokes teoremasidan foydalaning, bunda \(\vec F = \left( {3y{x^2} + {z^3}} \right)\,\vec i + {y^2}\,\vec j + 4y{x^2}\,\vec k\) va \(C\) uchburchaklar \(\left( {0,0,3} \right)\), \(\left( {0,2,0} \right)\) va \(\left( {4,0,0} \right)\). Agar siz uchburchak ustida bo'lsangiz va \(xy\) tekisligiga qarasangiz, \(C\) soat miliga teskari burilishga ega. Egri chiziqning eskizi uchun quyidagi rasmga qarang.
    A three dimensional sketch of the curve \(C\). It is the triangle with vertices \(\left( {0,0,3} \right)\) on the z-axis, \(\left( {0,2,0} \right)\) on the y-axis and \(\left( {4,0,0} \right)\) on the x-axis, and the arrows on each side show the direction it is traveled.
    Yechim