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Juliette [100K]
3 years ago
7

Ermine whether or not the vector field is conservative. if it is conservative, find a function f such that f = ∇f. (if the vecto

r field is not conservative, enter dne.) f(x, y, z) = 5y2z3 i + 10xyz3 j + 15xy2z2 k
Mathematics
1 answer:
anyanavicka [17]3 years ago
7 0
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.
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