
$$
curl {\bf F}=
\left \vert \matrix{ {\bf i} & {\bf j} & {\bf k}  \cr
{{\partial} \over {\partial x}} 
& {{\partial} \over {\partial y}} 
& {{\partial} \over {\partial z}} 
\cr
2e^{2x+3y+5z}+y^2+2y+1 & 3e^{2x+3y+5z}+2xy+3x+1 & 5e^{2x+3y+5z}
} \right \vert 
$$
 $$
={\bf i} \, 
\left ( {{\partial} \over {\partial y}} 5e^{2x+3y+5z} -
{{\partial} \over {\partial z}} (3e^{2x+3y+5z}+2xy+3x+1) \right )
$$
$$
-{\bf j} \, 
\left ( {{\partial} \over {\partial x}} 5e^{2x+3y+5z} -
{{\partial} \over {\partial z}} (2e^{2x+3y+5z}+y^2+2y+1) \right )
$$
$$
+{\bf k} \, 
\left ( {{\partial} \over {\partial x}} (3e^{2x+3y+5z}+2xy+3x+1) -
{{\partial} \over {\partial y}} (2e^{2x+3y+5z}+y^2+2y+1)   \right )
$$
$$
={\bf i} \, 
( (15e^{2x+3y+5z}- 15e^{2x+3y+5z})
-{\bf j} \, 
(10e^{2x+3y+5z} -10e^{2x+3y+5z} )
+{\bf k} \, 
(6e^{2x+3y+5z}+2y+3)-(6e^{2x+3y+5z}+2y+2)
$$
$$
={\bf i}\cdot 0  -{\bf j} \cdot + {\bf k}\cdot 1 \, = \, \langle 0,0,1 \rangle \quad .
$$
\end