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

We want to find a scalar function [tex]f(x,y,z)[/tex] whose gradient gives the vector function [tex]\nabla f(x,y,z)=\mathbf f(x,y,z)[/tex]. This means

[tex]\dfrac{\partial f}{\partial x}=5y^2z^3[/tex]
[tex]\dfrac{\partial f}{\partial y}=10xyz^3[/tex]
[tex]\dfrac{\partial f}{\partial z}=15xy^2z^2[/tex]

Integrating both sides of the first PDE with respect to [tex]x[/tex], we get

[tex]f(x,y,z)=5xy^2z^3+g(y,z)[/tex]

Differentiating with respect to [tex]y[/tex] gives

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

So we have

[tex]f(x,y,z)=5xy^2z^3+h(z)[/tex]

and differentiating with respect to [tex]z[/tex] gives

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

and so

[tex]f(x,y,z)=5xy^2z^3+C[/tex]

so [tex]\mathbf f[/tex] is indeed conservative.