Question

Determine whether or not the vector field is conservative. If it is, find a function f such that F = ∇f. If the vector field is not conservative, enter NONE. F(x, y, z) = 5y2z3 i + 10xyz3 j + 15xy2z2 k f(x, y, z) = + K

Answer 1

`\vec F` is conservative if we can find a scalar function
`f` such that
`\\abla f=\vec F`. This would require

`(\partial f)/(\partial x)=5y^2z^3`

`(\partial f)/(\partial y)=10xyz^3`

`(\partial f)/(\partial z)=15xy^2z^2`

Integrate both sides of the first PDE with respect to
`x`:

`f(x,y,z)=5xy^2z^3+g(y,z)` (*)

Differentiate both sides of (*) with respect to
`y`:

`(\partial f)/(\partial y)=10xyz^3=10xyz^3+(\partial g)/(\partial y)\implies(\partial g)/(\partial y)=0\implies g(y,z)=h(z)`

Differentiate both sides of (*) with respect to
`z`:

`(\partial f)/(\partial z)=15xy^2z^2=15xy^2z^2+(\mathrm dh)/(\mathrm dz)\implies(\mathrm dh)/(\mathrm dz)=0\implies h(z)=C`

So we have

`f(x,y,z)=5xy^2z^3+C`

and so
`\vec F` is indeed conservative.