Let F ( x , y , z ) = ⟨ x − 1 z , y − 1 z , ln ( x y ) ⟩ . Verify that F = ∇ f , where f ( x , y , z ) = z ln ( x y ) . Evaluate ∫ C F ⋅ d r , where r ( t ) = ⟨ e t , e 2 t , t 2 ⟩ for 1 ≤ t ≤ 3 . Evaluate ∫ C F ⋅ d r for any path C from P = ( 1 2 , 4 , 2 ) to Q = ( 2 , 2 , 3 ) contained in the region x > 0 , y > 0 . In part (c), why is it necessary to specify that the path lies in the region where x and y are positive?