Find the divergence of the vector field F

Question

Find the divergence of the vector field F= (-yz, xe^y,e^x + z²) using the definition div F V F. Show all steps in finding this dot product, do not use a shortcut formula.

Answer 1

div( F ) = ∇⋅ F = < ∂ / ∂x , ∂ / ∂y , ∂ / ∂z > ⋅ < -yz , xe^y , e^x + z² > = (∂ / ∂x )(-yz) +( ∂ / ∂y)(xe^y) +(∂ / ∂z)(e^x+z²) =

0 +xe^y +2z = xe^y +2z

Answer 2

div(F) = ∂F/∂x + ∂F/∂y + ∂F/∂z = 0 + xe^y + 2z = xe^y + 2z

Answer 3

div F = ∇•F = ∂F_x/∂x + ∂F_y/∂y + ∂F_z/∂z

F = -yzi +xe^yj +(e^x+z²)k

∂F_x/∂x = 0

∂F_y/∂y = xe^y

∂F_z/∂z = 2z

div F = xe^y + 2z