Question

$\mathbf{\text{∮}}\mathbf{(}\mathbf{yi}\mathbf{-}\mathbf{xj}\mathbf{+}\mathbf{zk}\mathbf{)}\mathbf{-}\mathbf{dr}$around the circumference of the circle of radius $\mathbf{2}$, center at the origin, in the $\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{}\mathbf{y}\mathbf{)}$plane.

Short answer

The solution derived is$\mathrm{I}=-8\mathrm{\pi }.$

Step-by-step solution

Given Information.

The given expression is$\text{∮}(\mathrm{yi}-\mathrm{xj}+\mathrm{zk})-\mathrm{dr}$

Definition of vector.

A quantity that has magnitude as well as direction is called a vector. It is typically denoted by an arrow in which the head determines the direction of the vector and the length determines its magnitude.

Apply Stokes’s theorem.

Apply Stokes' theorem and the fact that $\mathrm{\nabla }\times \mathrm{V}=-2\widehat{\mathrm{k}},\mathrm{Z}=0$and then the question can be solved as shown below.

$\mathrm{I}=-2{\int }_{\mathrm{\sigma }} \mathrm{d\sigma }$

$=2$(Area of circle)

$=-2\left(4\right)\mathrm{\pi }$

$=-8\mathrm{\pi }$

Hence, the solution derived is $\mathrm{I}=-8\mathrm{\pi }$.