Question

Ermine whether or not the vector field is conservative. if it is conservative, find a function f such that f = ∇f. (if the vector field is not conservative, enter dne.) f(x, y, z) = 5y2z3 i + 10xyz3 j + 15xy2z2 k

Answer 1

We want to find a scalar function
`f(x,y,z)` whose gradient gives the vector function
`\\abla f(x,y,z)=\mathbf f(x,y,z)`. This means


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

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

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

Integrating both sides of the first PDE with respect to
`x`, we get


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

Differentiating with respect to
`y` gives


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

So we have


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

and differentiating with respect to
`z` gives


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

and so


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

so
`\mathbf f` is indeed conservative.