Question

Given the vector field F(x,y,z)=(2xy + ycosx + 𝑥^2) i + ( 𝑥^2 + sinx +𝑦^2) j Determine whether F is conservative. If it is, find a potential function.

Answer 1

If F is conservative, we can find φ(x,y) such that
F = ∇φ
That means we need
∂φ/∂x = 2xy + y cosx + x^2
∂φ/∂y = x^2 + sinx + y^2
A little thought shows that the following works fine:
φ(x,y) = x^2 y + y sinx + 1/3 (x^3 + y^3)