Question

(1 point) Let F=−3yi+4xjF=−3yi+4xj. Use the tangential vector form of Green's Theorem to compute the circulation integral ∫CF⋅dr∫CF⋅dr where C is the positively oriented circle x2+y2=25x2+y2=25.

Answer 1

Solution: Given a vector function F:

Now, by using the tangential form of the green's theorem, we've to calculate the circulation given by:

Here C is positively oriented curve given by:

To do this, we should know the concept of Green's theorem (tangential-form).

Green's theorem: Let C be a piecewise smooth, simple closed curve in the plane and D be the region inside the enclosed by the curve C. If F is a vector field such that

Where M and N are functions of x and y and having continuous partial derivatives in the region D, then according to the green's theorem:

  ...(1)

Here, the integral is transversed in the counter-clockwise direction

Now, in our case, the vector function and curve C is given by:

Since D is the region enclosed by the curve C, therefore, it will be:

We've to calculate:

Since

Therefore,

Now, applying the tangential form of the green's theorem, we'll get:

In the graph of the region D above, we can see that the value of x and y varies between:

Therefore, our integration becomes:

Using standard integration:

We'll get:

Using standard integration:

We'll get:

​​​​​​​I hope it helps you!