Answer
$\nabla g(x,y,z)=-[\dfrac{2xi}{(x^2+y^2)}+ \dfrac{2yj}{(x^2+y^2)}]+e^z k$
Work Step by Step
The gradient field of $f(x,y,z)$ can be expressed as:
$\nabla f(x,y,z) =\dfrac{\partial (f(x,y,z))}{\partial x} i+\dfrac{\partial (f(x,y,z))}{\partial y} j+\dfrac{\partial (f(x,y,z))}{\partial z}k $
Since, $ g(x,y,z)=e^{z}-\ln (x^2+y^2)$
Therefore, the gradient field can be computed as
$\nabla g(x,y,z)=e^{z}-\ln (x^2+y^2)=\dfrac{1}{2} [\dfrac{2x}{(x^2+y^2+z^2)}i +\dfrac{2y}{(x^2+y^2+z^2)} j +\dfrac{2z}{(x^2+y^2+z^2)} k ] \\ =-\dfrac{2xi}{(x^2+y^2)}+ [-\dfrac{2yj}{(x^2+y^2)}]+e^z k \\ =-[\dfrac{2xi}{(x^2+y^2)}+ \dfrac{2yj}{(x^2+y^2)}]+e^z k$
