Answer
$\nabla \phi (x, y)=\lt e^{-z}, \cos (x+y) , e^{-z}, \cos (x+y), -e^{-z}, \sin (x+y) \gt$
Work Step by Step
Our aim is to compute the the gradient vector field vector.
Here, we have $\phi (x, y)=e^{-z} \sin (x+y)$
Now, $\nabla \phi (x, y)=\lt \dfrac{\partial \phi (x, y)}{\partial x} , \dfrac{\partial \phi (x, y)}{\partial y} , \dfrac{\partial \phi (x, y)}{\partial z} \gt $
or, $\nabla \phi (x, y)=\lt \dfrac{\partial [e^{-z} \sin (x+y)]}{\partial x} , \dfrac{\partial [e^{-z} \sin (x+y)]}{\partial y}, \dfrac{\partial [e^{-z} \sin (x+y)]}{\partial z} \gt$
or, $= \lt e^{-z} \cos (x+y) \dfrac{\partial (x+y)}{\partial x} , e^{-z} \cos (x+y) \dfrac{\partial (x+y)}{\partial y}, e^{-z} \cos (x+y) \dfrac{\partial (x+y)}{\partial z} \gt $
Thus, we have $\nabla \phi (x, y)=\lt e^{-z}, \cos (x+y) , e^{-z}, \cos (x+y), -e^{-z}, \sin (x+y) \gt$
