Question

Найти градиент функции u=f(x,y,z) в точке М u = x^2*y-корень(x*y+z^2) M(1,5,-2).

Answer 1

$u=x^2y-\sqrt{xy+z^2}\ \ ,\ \ \ M(1;5;-2)\\\\u'_{x}=2xy-\dfrac{y}{2\sqrt{xy+z^2}}\ \ ,\ \ \ u'_{y}=x^2-\dfrac{x}{2\sqrt{xy+z^2}}\ \ ,\ \ u'_{z}=\dfrac{-2z}{2\sqrt{xy+z^2}}\\\\\\u'_{x}\Big|_{M}=10-\dfrac{5}{\sqrt{9}}=10-\dfrac{5}{3}=\dfrac{25}{3}\ \ ,\ \ u'_{y}\Big|_{M}=25-\dfrac{1}{3}=\dfrac{74}{3}\ \ ,\ \ u'_{z}\Big|_{M}=\dfrac{2}{3}\\\\\\\overline {grad\, u}\Big|_{M}=\dfrac{25}{3}\, \vec{i}+\dfrac{74}{3}\, \vec{j}+\dfrac{2}{3}\, \vec{k}$