Question

Determine 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) = i + sin(z)j + y cos(z)k.

Answer 1

If F(x, y, z) = i + sin(z) j + y cos(z) k is conservative, then there exists a scalar function f(x, y, z) such that grad(f) = F, which means

∂f/∂x = 1

∂f/∂y = sin(z)

∂f/∂z = y cos(z)

Integrating each each of these equations gives

∫ ∂f/∂x dx = ∫ dx   ⇒   f(x, y, z) = x + α(y, z)

∫ ∂f/∂y dy = ∫ sin(z) dy   ⇒   f(x, y, z) = y sin(z) + β(x, z)

∫ ∂f/∂z dx = ∫ y cos(z) dz   ⇒   f(x, y, z) = y sin(z) + γ(x, y)

It follows that α(y, z) = y sin(z) and β(x, z) + γ(x, y) = x + C where C is a constant. So

f(x, y, z) = x + y sin(z) + C

and F is indeed conservative.