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
`\\abla 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.