Question

Let F=-1yi+1xj. Use the tangential vector form of Green's Theorem to compute the circulation integral CF*dr where C is the positively oriented circle x2+y2=1.

Answer 1

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

Green's Theorem

Let C be a smooth curve given by a vector function r(t),a≤t≤b and F=Pi+Qj be a force field on R2. The circulation of the force field F along C is given by the line integral of F along C, ∫CF⋅dr=∫baF(r(t))⋅r′(t)dt.

The tangential form of Green's Theorem connects circulation integrals along a simple closed curve C counterclockwise oriented with double integrals over the plane region D bounded by C,

more precisely, it gives the following equation, ∫CF⋅dr=∫CPdx+Qdy=∬D(∂Q∂x−∂P∂y)dA

Answer and Explanation:

Let D be the disk x2+y2≤1 bounded

Answer 2

Let C be a smooth curve given by a vector function r(t),a≤t≤br(t),a≤t≤b and F=Pi+QjF=Pi+Qjbe a force field on R2R2. The circulation of the force field F along C is given by the line integral of F along C, ∫CF⋅dr=∫baF(r(t))⋅r′(t)dt∫CF⋅dr=∫abF(r(t))⋅r′(t)dt.

The tangential form of Green's Theorem connects circulation integrals along a simple closed curve C counterclockwise oriented with double integrals over the plane region D bounded by C,

more precisely, it gives the following equation, ∫CF⋅dr=∫CPdx+Qdy=∬D(∂Q∂x−∂P∂y)dA∫CF⋅dr=∫CPdx+Qdy=∬D(∂Q∂x−∂P∂y)dA

Answer and Explanation:

Let D be the disk x2+y2≤1x2+y2≤1 bounded by C, which is counterclockwise oriented.

We have...

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