Задача 17 (Знайти похідну скалярного поля в точці)

Знайти похідну скалярного поля U в точці М0, що належить заданій кривій, за напрямом цієї кривої.

 U=\frac{x}{y}-\frac{y}{x}

 M_{0}(1;1)\in x^{2}+y^{2}-2y=0

♦  \frac{\partial U(M_{0}}{\partial l}=U_{x}^{'}\left(M_{0} \right)cos\alpha +U_{y}^{'}(M_{0})cos\beta =

 U_{x}^{'}=\frac{1}{y}+\frac{y}{x^{2}}

 U_{x}^{'}(M_{0}) = \frac{1}{1}+\frac{1}{1} = 1+1 = 2

  U_{y}^{'} = - \frac{x}{y^{2}} - \frac{1}{x}

 U_{y}^{'}(M_{0})=-\frac{1}{1}-\frac{1}{1}=-1-1=-2

 f(x;y):\; x^{2}+y^{2}-2y=0

 \begin{matrix}  f_{x}^{'} = 2x & f_{x}^{'}(M_{0}) = 2 \\ f_{y}^{'} = 2y-2 & f_{y}^{'}(M_{0}) = 0 \end{matrix}

 \vec{n}=\left\{f_{x}^{'}; f_{y}^{'} \right\}=\left\{2;0 \right\}

 \left|\vec{n} \right|=\sqrt{4+0}=2

  cos\alpha =\frac{2}{2}=1 \; cos\beta =\frac{0}{2}=0

 =2\cdot 1-2\cdot 0=2-0=2 .

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