Conservative Vector Fields Practice

F = ⟨ x y e^{z}, 0, yz e^{x} ⟩

d i v (F) = \nabla \cdot F = \frac{\partial}{\partial x} x y e^{z} + \frac{\partial}{\partial y} 0 + \frac{\partial}{\partial z} yz e^{x} = y e^{z} y e^{x}

c u r l (F) = \nabla \times F = i \frac{\partial}{\partial x} x y e^{z} j \frac{\partial}{\partial y} 0 k \frac{\partial}{\partial z} yz e^{x} = ⟨ z e^{y}, x y e^{x}, - x e^{z} ⟩

f_{x} f_{y} f_{z} = yz sin x y \to f = - z cos x y + g (y, z) = x z sin x y \to f = - z cos x y + h (x, z) = - cos (x y) \to f = - z cos x y + k (x, y)

Problem 9

S A : z = x^{3} + y^{2}, b e tw ee n y = 5 x an d y = x^{2}

S A = \int \int_{D} 1 + z_{x}^{2} + z y^{2}