Question

$\mathbf{\oint }\mathbf{2}\mathit{x}\mathit{d}\mathit{y}\mathbf{-}\mathbf{3}\mathit{y}\mathit{d}\mathit{x}$around the square with vertices$\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{2}\mathbf{)}\mathbf{,}\mathbf{(}\mathbf{2}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{,}\mathbf{(}\mathbf{-}\mathbf{2}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{,}\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{-}\mathbf{2}\mathbf{)}\mathbf{.}\mathbf{}$

Short answer

The integral will give the solution to be 40.

Step-by-step solution

Given information

The vertices of the square are $(0,2),(2,0),(-2,0,(0,-2)$

Definition of conservative force and scalar potential

A force is said to be conservative if $\mathbf{\nabla }\mathbf{\times }\mathit{F}\mathbf{=}\mathbf{0}\mathbf{.}$.

The scalar potential is independent of the path. The scalar potential is the sum of potential in all the 3 dimensions calculated separately.

The formula for the scalar potential is $\mathit{W}\mathbf{=}\mathbf{\int }\mathit{F}\mathbf{.}\mathit{d}\mathit{r}\mathbf{.}$.

Use substitution.

The green theorem in a plane is mentioned.

$\iint \frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})=\oint Pdx+Qdy$

Takethe values as mentioned below.

$$\begin{gathered} P=-3y \\ Q=2x \end{gathered}$$

Substitute the value.

$$\begin{gathered} \frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}=2+3 \\ \frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}=5 \end{gathered}$$

Use Greens Theorem.

Use the Green theorem.

$$\begin{gathered} -3xdx+2ydy \\ 5\iint dxdy \\ 5{\int }_0^{\sqrt{8}}{\int }_0^{\sqrt{8}}dxdy \\ 5{\int }_0^{\sqrt{8}}\sqrt{8}dx \\ 5\sqrt{8}\sqrt{8} \\ 40 \end{gathered}$$

Hence the integral $\oint 2xdy-3ydx$ gives 40