Q. 39 - Calculus

Question

Show that the vector fields in Exercises 33–40 are not conservative.

$\mathbf{G}(x,y,z)=(3,yz,z+12)$

Step-by-Step Solution

Answer

The vector fields is not conservative because $\frac{\partial F_1}{\partial z}\ne \frac{\partial F_2}{\partial y}.$. 

Step 1. Given Information

We have to show that the vector fields in the given exercise is not conservative.
$\mathbf{G}(x,y,z)=(3,yz,z+12)$

Step 2. A vector field G ( x , y ) = ( G 1 ( x , y ) , G 2 ( x , y ) ) is not conservative if and only if ∂ F 1 ∂ z ≠ ∂ F 2 ∂ y .

For the vector field $\mathbf{G}(x,y,z)=(3,yz,z+12)$

$$\begin{gathered} \frac{\partial F_1}{\partial z}=\frac{\partial }{\partial z}yz \\ \frac{\partial F_1}{\partial z}=y\frac{\partial }{\partial z}z \\ \frac{\partial F_1}{\partial z}=y \end{gathered}$$

Step 3. Now finding ∂ F 2 ∂ y

$$\begin{gathered} \frac{\partial F_2}{\partial y}=\frac{\partial }{\partial y}(z+12) \\ \frac{\partial F_2}{\partial y}=\frac{\partial }{\partial y}z+\frac{\partial }{\partial y}12 \\ \frac{\partial F_2}{\partial y}=0 \end{gathered}$$

Hence, $\frac{\partial F_1}{\partial z}\ne \frac{\partial F_2}{\partial y}.$