यदि $u = \log_e(x^2 + y^2) + \tan^{-1}\left(\frac{y}{x}\right)$ है,तो $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = $

यदि $u = \tan^{-1}(x + y)$ है,तो $x\frac{\partial u}{\partial x} + y\frac{\partial u}{\partial y} = $

यदि ${z^2} = \frac{{x^{1/2} + y^{1/2}}}{{x^{1/3} + y^{1/3}}}$ है,तो $x\frac{{\partial z}}{{\partial x}} + y\frac{{\partial z}}{{\partial y}} = $

यदि $u = (x^2 + y^2 + z^2)^{3/2}$ है,तो $\left( \frac{\partial u}{\partial x} \right)^2 + \left( \frac{\partial u}{\partial y} \right)^2 + \left( \frac{\partial u}{\partial z} \right)^2 = $

यदि $z = \sin^{-1}\left( \frac{x+y}{\sqrt{x} + \sqrt{y}} \right)$ है,तो $x\frac{\partial z}{\partial x} + y\frac{\partial z}{\partial y}$ का मान ज्ञात कीजिए।

यदि $u \equiv u(x, y) = \sin(y + ax) - (y + ax)^2$ है,तो यह क्या दर्शाता है?