Question

Determine whether or not the vector field is conservative. If it is, find a function f such that F=▽f.F(x,y,z)=3y2z3i+6xyz3j+9xy2z2kf(x,y,z)=+K

Answer 1

If $\vec F$ is conservative, then there is a scalar function $f$ such that $\nabla f=\vec F$. This means

$f_x=3y^2z^3$

$f_y=6xyz^3$

$f_z=9xy^2z^2$

Integrate both sides of the first PDE wrt $x$:

$f(x,y,z)=3xy^2z^3+g(y,z)$

Differentiate wrt $y$:

$f_y=6xyz^3=6xyz^3+g_y\implies g_y=0\implies g(y,z)=h(z)$

Differentiate wrt $z$:

$f_z=9xy^2z^2=9xy^2z^2+g_z=9xy^2z^2+h_z\implies h_z=0\implies h(z)=C$

Then

$f(x,y,z)=3xy^2z^3+C$

so $\vec F$ is conservative.