Question

If $u=\left|\begin{array}{ccc}1& 1& 1\\ x& y& z\\ x^2& y^2& z^2\end{array}\right|$then $x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}+z\frac{\partial u}{\partial z}=$

• A. $u$
• B. $2u$
• C. $3u$
• D. $4u$

Answer 1

The correct option is C $3u$

Given $u=\left|\begin{array}{ccc}1& 1& 1\\ x& y& z\\ x^2& y^2& z^2\end{array}\right|=y{z}^2-y^{2}z+x^{2}z-x{z}^2+x{y}^2-x^{2}y$

$\frac{du}{dx}=2xz-z^2+y^2-2xy$

$\frac{du}{dy}=z^2-2yz+2xy-x^2$

$\frac{du}{dz}=2yz-y^2+x^2-2xz$

$x\frac{du}{dx}+y\frac{du}{dy}+z\frac{du}{dz}=3(y{z}^2-y^{2}z+x^{2}z-x{z}^2+x{y}^2-x^{2}y)=3u$