Question

Решить уравнение функции нескольких переменных

Answer 1

$f(x;y)=arctg\dfrac{1}{x^2+y^2}\\\\\\\dfrac{\partial f}{\partial x}=\dfrac{1}{1+\frac{1}{(x^2+y^2)^2}}\cdot \dfrac{-2x}{(x^2+y^2)^2}=\dfrac{-2x(x^2+y^2)^2}{((x^2+y^2)^2+1)(x^2+y^2)^2}=\\\\\\=\dfrac{-2x}{(x^2+y^2)^2+1}\\\\\\\dfrac{\partial f}{\partial y}=\dfrac{-2y}{(x^2+y^2)^2+1}\\\\\\d_{x}f=\dfrac{\partial f}{\partial x}\cdot dx=\dfrac{-2y}{(x^2+y^2)^2+1}\cdot dx\\\\\\d_{y}f=\dfrac{\partial f}{\partial x}\cdot dy=\dfrac{-2y}{(x^2+y^2)^2+1}\cdot dy$

$dz=-\dfrac{2x}{(x^2+y^2)^2+1}\cdot dx-\dfrac{2y}{(x^2+y^2)^2+1}\cdot dy$