2.2.72 Derivatives and differentials

condition: 1. Find the domain of definition of a function of two variables \( z=f(x, y) \). Draw it on the coordinate plane and shade it. 2. Check whether the function of two variables \( \quad z=f(x, y) \quad \) satisfies the specified differential equation. 1) \( z=\sqrt{1-x^{2}-y^{2}} \) 2) a) \( z=x \cdot \sin \left(x^{2}-y^{2}\right), \quad x^{2} \cdot \frac{\partial z}{\partial y}+x y \frac{\partial z}{\partial x}=z y \), b) \( z=\cos ^{2}(x+y)+\ln (x-y), \frac{\partial^{2} z}{\partial x^{2}}=\frac{\partial^{2} z}{\partial y^{2}} \).