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 $\nabla f(x,y,z)=\mathbf f(x,y,z)$. This means

$\dfrac{\partial f}{\partial x}=5y^2z^3$
$\dfrac{\partial f}{\partial y}=10xyz^3$
$\dfrac{\partial f}{\partial z}=15xy^2z^2$

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

$10xyz^3=10xyz^3+\dfrac{\partial g}{\partial y}\implies\dfrac{\partial g}{\partial y}=0\implies g(y,z)=h(z)$

So we have

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

and differentiating with respect to $z$ gives

$15xy^2z^2=15xy^2z^2+\dfrac{\mathrm dh}{\mathrm dz}\implies\dfrac{\mathrm dh}{\mathrm dz}=0\implies h(z)=C$

and so

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

so $\mathbf f$ is indeed conservative.