Evaluate the circulation of g⃗ =xyi⃗ +zj⃗ +3yk⃗ around a square of side length 9, centered at the origin, lying in the yz-plane, and oriented counterclockwise when viewed from the positive x-axis.

We have,the circulation of    around a square of side 3, centered at the origin, lying in the yz-plane, and oriented counterclockwise when viewed from the positive x-axis is

From the Question we have the equation to be

Therefore

Where

For i

For j

For z

Therefore

Generally considering as the origin

We apply Stoke's Theorem

Therefore

The circulation of around a square of side 3, centered at the origin, lying in the yz-plane, and oriented counterclockwise when viewed from the positive x-axis is

Given that

has a fairly simple curl,

we can take advantage of Stokes' theorem by transforming the line integral of along the boundary of the square (call it ) to the integral of over itself. Parameterize by

with and . Then take the normal vector to to be

so that