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16.7-bo'lim : Grin teoremasi

  1. \( \displaystyle \int\limits_{C}{{\left( {y{x^2} - y} \right)\,dx + \left( {{x^3} + 4} \right)\,dy}}\) ni baholash uchun Grin teoremasidan foydalaning, bu erda \(C\) quyida ko'rsatilgan.
    The curve is a triangle with vertices (0,0), (6,3) and (-3,3). It is traced out with a counter clockwise rotation.
  2. \( \displaystyle \int\limits_{C}{{\left( {7x + {y^2}} \right)dy - \left( {{x^2} - 2y} \right)\,dx}}\) ni baholash uchun Green teoremasidan foydalaning, bu erda \(C\) quyida ko'rsatilgandek ikkita doiradir.
    There are two circles, both centered at the origin, in the sketch. The first has a radius of 1 and a counter clockwise rotation. The second has a radius of 3 and a clockwise rotation.
  3. \( \displaystyle \int\limits_{C}{{\left( {{y^2} - 6y} \right)\,dx + \left( {{y^3} + 10{y^2}} \right)\,dy}}\) ni baholash uchun Grin teoremasidan foydalaning, bu erda \(C\) quyida ko'rsatilgan.
    There are four parts to this curve. The first part is the portion of $y=x^{2]-3$ starting at (-1,-2) and ending at (1,-2). This is followed by a line starting at (1,-2) and ending at (1,1). The next part is the portion of $y=2-x^{2}$ starting at (1,1) and ending at (-1,1). The final part is a line starting at (-1,1) and ending at (-1,-2).
  4. Quyida \(C\) ko'rsatilgan \( \displaystyle \int\limits_{C}{{x{y^2}\,dx + \left( {1 - x{y^3}} \right)\,dy}}\) ni baholash uchun Green teoremasidan foydalaning.
    The curve is a diamond shaped curve with vertices (-1,-1), (1,1), (1,4) and (-1,2). It is traced out with a counter clockwise rotation.
  5. \( \displaystyle \oint_{C}{{\left( {{y^2} - 4x} \right)\,dx - \left( {2 + {x^2}{y^2}} \right)\,dy}}\) ni baholash uchun Grin teoremasidan foydalaning, bu erda \(C\) quyida ko'rsatilgan.
    The curve is a triangle with vertices (0,0), (5,10) and (5,0). It is traced out with a clockwise rotation.
  6. \( \displaystyle \oint_{C}{{\left( {{y^3} - x{y^2}} \right)\,dx + \left( {2 - {x^3}} \right)\,dy}}\) ni baholash uchun Grin teoremasidan foydalaning, bu erda \(C\) quyida ko'rsatilgan.
    This curve start with the portion of $x^{2}+y^{2}=16$ starting at (0,-4) and ending at (4,0). This is followed by a line starting at (4,0) and ending at the origin. The final portion of the curve is a line starting at the origin and ending at (0,-4).
  7. \(C\) quyida ko'rsatilgan \( \displaystyle \oint_{C}{{\left( {6 + {x^2}} \right)\,dx + \left( {1 - 2xy} \right)\,dy}}\) uchun Green teoremasini tasdiqlang (a) chiziqli integralni to'g'ridan-to'g'ri hisoblash va (b) chiziqli integralni hisoblash uchun Grin teoremasidan foydalanish.
    This curve start with the portion of $x^{2}+y^{2}=1$ starting at (1,0) and ending at (-1,0). This is followed by a line starting at (-1,0) and ending at (1,0).
  8. \(C\) quyida ko'rsatilgan \( \displaystyle \oint_{C}{{\left( {6y - 3{y^2} + x} \right)\,dx + y{x^3}dy}}\) uchun Green teoremasini tasdiqlang (a) chiziqli integralni to'g'ridan-to'g'ri hisoblash va (b) chiziqli integralni hisoblash uchun Grin teoremasidan foydalanish.
    The curve is a triangle with vertices (0,0), (1,3) and (1,5). It is traced out with a clockwise rotation.