Suppose that the plane region D, its boundary curve C, and the functions P and Q satisfy the hypothesis of Green's Theorem. Considering the vector field F = Pi+Qj, prove the vector form of Green's Theorem oint_C F*n ds = int int_D div F(x,y) dA where n(t) is the outward unit normal vector to C.

illusiia 2020-11-02 Answered
Suppose that the plane region D, its boundary curve C, and the functions P and Q satisfy the hypothesis of Greens
You can still ask an expert for help

Want to know more about Green's, Stokes', and the divergence theorem?

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

Luvottoq
Answered 2020-11-03 Author has 95 answers
Step 1
We have to prove that,
CF.ndsintD÷F(x,y)dA
Step 2
Consider Vector field ,
F=Pi+Qj
÷(F)=Px+Qy
Step 3
If C is given by the vector equation ,
r(t)=x(t)i+y(t)j atb
Then unit tangent vector is
T(t)=x(t)|r|(r(t))i+y(t)|r(t)|j
n(t) is the outward unit normal vector C is given by,
n(t)=y(t)|r|(r(t))i+x(t)|r(t)|j
As we know
Step 4
CF.nds=ab(F.n)(t)|r(t)|dt
=ab[P(x(t),y(t))y(t)|r(t)|Q(x(t),y(t))x(t)|r(t)|]|r(t)|dt
=abP(x(t),y(t))y(t)dtQ(x(t),y(t))x(t)dt
=CPdyQdx
=D(Px+Qy)dA
Put Div(F) value in the above equation
Step 5
CF.nds=DDiv(F)dA

We have step-by-step solutions for your answer!

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

New questions