Q. 36 - Calculus

Question

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

$\mathbf{G}(x,y)=y\mathbf{i}-x\mathbf{j}$

Step-by-Step Solution

Answer

The vector fields is not conservative because $\frac{\partial G_1}{\partial y}\ne \frac{\partial G_2}{\partial x}$. 

Step 1. Given Information

We have to show that the vector fields in the given exercise is not conservative.
$\mathbf{G}(x,y)=y\mathbf{i}-x\mathbf{j}$

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

For the vector field 

$$\begin{gathered} \frac{\partial F_1}{\partial y}=\frac{\partial }{\partial y}y \\ \frac{\partial F_1}{\partial y}=1 \end{gathered}$$

Step 3. Now finding ∂ F 2 ∂ y

$$\begin{gathered} \frac{\partial F_2}{\partial x}=\frac{\partial }{\partial x}(-x) \\ \frac{\partial F_2}{\partial x}=-1 \end{gathered}$$

Hence, $\frac{\partial G_1}{\partial y}\ne \frac{\partial G_2}{\partial x}.$