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16.7-bo'lim: Grin teoremasi

  1. \( \displaystyle \int\limits_{C}{{y{x^2}\,dx - {x^2}\,dy}}\) ni baholash uchun Grin teoremasidan foydalaning, bu erda \(C\) quyida ko'rsatilgan.
    The curve \(C\). It is a closed curve made up of the left half of the circle \({x^2} + {y^2} = 25\) together with the portion of the y-axis from \(\left( {0, - 5} \right)\) to \(\left( {0,5} \right)\). The arrows show that the curve starts at \(\left( {0,5} \right)\), travels around the semicircle through \(\left( { - 5,0} \right)\) to \(\left( {0, - 5} \right)\) and then back up the y-axis to the starting point.
    Yechim
  2. \( \displaystyle \int\limits_{C}{{\left( {6y - 9x} \right)dy - \left( {yx - {x^3}} \right)\,dx}}\) ni baholash uchun Grin teoremasidan foydalaning, bu erda \(C\) quyida ko'rsatilgan.
    The curve \(C\). It is a closed four sided figure with corners at \(\left( { - 1,4} \right)\), \(\left( { - 1, - 1} \right)\), \(\left( {1, - 1} \right)\) and \(\left( {1,2} \right)\). The arrows show that it is traveled in the counter clockwise direction, going down the left side, right along the bottom, up the right side and then back up along the slanted top side to \(\left( { - 1,4} \right)\).
    Yechim
  3. \( \displaystyle \int\limits_{C}{{{x^2}{y^2}\,dx + \left( {y{x^3} + {y^2}} \right)\,dy}}\) ni baholash uchun Grin teoremasidan foydalaning, bu erda \(C\) quyida ko'rsatilgan.
    The curve \(C\). It is a closed triangle with corners at the origin, \(\left( {4,2} \right)\) and \(\left( {4, - 8} \right)\). The arrows show that it is traveled in the clockwise direction, going from the origin up to \(\left( {4,2} \right)\), straight down the right side to \(\left( {4, - 8} \right)\) and then back up to the origin.
    Yechim
  4. \( \displaystyle \int\limits_{C}{{\left( {{y^4} - 2y} \right)\,dx - \left( {6x - 4x{y^3}} \right)\,dy}}\) ni baholash uchun Green teoremasidan foydalaning, bu erda \(C\) quyida ko'rsatilgan.
    The curve \(C\). It is the closed rectangle with corners at the origin, \(\left( {6,0} \right)\), \(\left( {6,4} \right)\) and \(\left( {0,4} \right)\). The arrows show that it is traveled in the counter clockwise direction, going right along the bottom, up the right side, left across the top and back down the left side.
    Yechim
  5. \( \displaystyle \oint_{C}{{\left( {x{y^2} + {x^2}} \right)\,dx + \left( {4x - 1} \right)\,dy}}\) uchun Green teoremasini tasdiqlang, bu erda \(C\) quyida ko'rsatilgan. (a)chiziqli integralni to'g'ridan-to'g'ri hisoblash va (b) chiziqli integralni hisoblash uchun Grin teoremasidan foydalanish.
    The curve \(C\). It is the closed triangle with corners at \(\left( { - 3,0} \right)\), the origin and \(\left( {0,3} \right)\). The arrows show that it is traveled in the counter clockwise direction, going right along the x-axis to the origin, up the y-axis to \(\left( {0,3} \right)\) and then back down the slanted side to \(\left( { - 3,0} \right)\).
    Yechim