Question

True or false:

There is a function  $z$  whose partial derivatives are $\displaystyle{\frac{\partial z}{\partial y}}=x$ and  $\displaystyle{\frac{\partial z}{\partial x}}=-y$

Answer 1

Since, the partial derivative of z with respect to y is iota (which is not in real), we can not get any complex number while differentiate a real valued function.

$\space \frac{\partial z}{\partial y}=i.\\ \text{Therefore, the given statement is false.}$