# 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.

Question
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 $$\displaystyle\oint_{{C}}{F}\cdot{n}{d}{s}=\int\int_{{D}}\div{F}{\left({x},{y}\right)}{d}{A}$$
where n(t) is the outward unit normal vector to C.

2020-11-03
Step 1
We have to prove that,
$$\displaystyle\oint_{{C}}{F}.{n}{d}{\sin{{t}}}\int_{{D}}\div{F}{\left({x},{y}\right)}{d}{A}$$
Step 2
Consider Vector field ,
F=Pi+Qj
$$\displaystyle\div{\left({F}\right)}=\frac{{\partial{P}}}{{\partial{x}}}+\frac{{\partial{Q}}}{{\partial{y}}}$$
Step 3
If C is given by the vector equation ,
r(t)=x(t)i+y(t)j $$\displaystyle{a}\le{t}\le{b}$$
Then unit tangent vector is
$$\displaystyle{T}{\left({t}\right)}=\frac{{{x}'{\left({t}\right)}}}{{{\left|{r}\right|}{\left({r}'{\left({t}\right)}\right)}}}{i}+\frac{{{y}'{\left({t}\right)}}}{{{\left|{{r}'{\left({t}\right)}}\right|}}}{j}$$
n(t) is the outward unit normal vector C is given by,
$$\displaystyle{n}{\left({t}\right)}=\frac{{{y}'{\left({t}\right)}}}{{{\left|{r}\right|}{\left({r}'{\left({t}\right)}\right)}}}{i}+\frac{{{x}'{\left({t}\right)}}}{{{\left|{{r}'{\left({t}\right)}}\right|}}}{j}$$
As we know
Step 4
$$\displaystyle\oint_{{C}}{F}.{n}{d}{s}={\int_{{a}}^{{b}}}{\left({F}.{n}\right)}{\left({t}\right)}{\left|{{r}'{\left({t}\right)}}\right|}{\left.{d}{t}\right.}$$
$$\displaystyle={\int_{{a}}^{{b}}}{\left[\frac{{{P}{\left({x}{\left({t}\right)},{y}{\left({t}\right)}\right)}{y}'{\left({t}\right)}}}{{{\left|{{r}'{\left({t}\right)}}\right|}}}-\frac{{{Q}{\left({x}{\left({t}\right)},{y}{\left({t}\right)}\right)}{x}'{\left({t}\right)}}}{{{\left|{{r}'{\left({t}\right)}}\right|}}}\right]}{\left|{{r}'{\left({t}\right)}}\right|}{\left.{d}{t}\right.}$$
$$\displaystyle={\int_{{a}}^{{b}}}{P}{\left({x}{\left({t}\right)},{y}{\left({t}\right)}\right)}{y}'{\left({t}\right)}{\left.{d}{t}\right.}-{Q}{\left({x}{\left({t}\right)},{y}{\left({t}\right)}\right)}{x}'{\left({t}\right)}{\left.{d}{t}\right.}$$
$$\displaystyle=\int_{{C}}{P}{\left.{d}{y}\right.}-{Q}{\left.{d}{x}\right.}$$
$$\displaystyle=\int\int_{{D}}{\left(\frac{{\partial{P}}}{{\partial{x}}}+\frac{{\partial{Q}}}{{\partial{y}}}\right)}{d}{A}$$
Put Div(F) value in the above equation
Step 5
$$\displaystyle\oint_{{C}}{F}.{n}{d}{s}=\int\int_{{D}}{D}{i}{v}{\left({F}\right)}{d}{A}$$

### Relevant Questions

Use Green's Theorem in the form of this equation to prove Green's first identity, where D and C satisfy the hypothesis of Green's Theorem and the appropriate partial derivatives of f and g exist and are continuous. (The quantity grad g · n = Dng occurs in the line integral. This is the directional derivative in the direction of the normal vector n and is called the normal derivative of g.)
$$\displaystyle\oint_{{c}}{F}\cdot{n}{d}{s}=\int\int_{{D}}\div{F}{\left({x},{y}\right)}{d}{A}$$
Use Green’s Theorem to evaluate around the boundary curve C of the region R, where R is the triangle formed by the point (0, 0), (1, 1) and (1, 3).
Find the work done by the force field F(x,y)=4yi+2xj in moving a particle along a circle $$\displaystyle{x}^{{2}}+{y}^{{2}}={1}$$ from(0,1)to(1,0).
Let C be the ellipse contained in the xy plane whose equation is $$\displaystyle{4}{x}^{{2}}+{y}^{{2}}={4}$$, oriented clockwise. The force field F described by $$\displaystyle{F}{\left({x},{y},{z}\right)}={x}^{{2}}{i}+{2}{x}{j}+{z}^{{2}}{k}$$, moves a particle along C in the same direction as the curve orientation, performing a W job. C as the surface boundary S: $$\displaystyle{z}={4}-{4}{x}^{{2}}-{y}^{{2}},{z}\ge{0}$$ (with ascending orientation, that is, the component in the z direction equal to 1) and assuming $$\displaystyle\pi={3.14}$$, we can state what:
a) It is not necessary to apply Stokes' Theorem, as C is a closed curve and therefore W = 0.
b) Inverting the orientation of the surface S, we can apply Stokes' Theorem and conclude that W = 12.56.
c) We can apply Stokes' Theorem and conclude that W = 6.28
d) We can apply Stokes' Theorem and conclude that W = 12.56.
Using the Divergence Theorem, evaluate $$\displaystyle\int\int_{{S}}{F}.{N}{d}{S}$$, where $$\displaystyle{F}{\left({x},{y},{z}\right)}={\left({z}^{{3}}{i}-{x}^{{3}}{j}+{y}^{{3}}{k}\right)}$$ and S is the sphere $$\displaystyle{x}^{{2}}+{y}^{{2}}+{z}^{{2}}={a}^{{2}}$$, with outward unit normal vector N.
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The work required to move an object around a closed curve C in the presence of a vector force field is the circulation of the force field on the curve.
b. If a vector field has zero divergence throughout a region (on which the conditions of Green’s Theorem are met), then the circulation on the boundary of that region is zero.
c. If the two-dimensional curl of a vector field is positive throughout a region (on which the conditions of Green’s Theorem are met), then the circulation on the boundary of that region is positive (assuming counterclockwise orientation).
Flux integrals Compute the outward flux of the following vector fields across the given surfaces S. You should decide which integral of the Divergence Theorem to use.
$$\displaystyle{F}={\left\langle{x}{\sin{{y}}},-{\cos{{y}}},{z}{\sin{{y}}}\right\rangle}$$ , S is the boundary of the region bounded by the planes x = 1, y = 0, $$\displaystyle{y}=\frac{\pi}{{2}},{z}={0}$$, and z = x.
Let $$\displaystyle{F}{\left({x},{y}\right)}={\left\langle{5}{\cos{{\left({y}\right)}}},{8}{\sin{{\left({y}\right)}}}\right\rangle}$$.Compute the flux $$\displaystyle\oint{F}\cdot{n}{d}{s}$$ of F across the boundary of the rectangle $$\displaystyle{0}\le{x}\le{5},{0}\le{y}\le\frac{\pi}{{2}}$$ using the vector form of Green's Theorem.
$$\displaystyle\oint{F}\cdot{n}{d}{s}=$$ ?
$$\displaystyle{F}={\left\langle{z}-{x},{x}-{y},{2}{y}-{z}\right\rangle}$$, D is the region between the spheres of radius 2 and 4 centered at the origin.
Let $$\displaystyle{F}{\left({x},{y}\right)}={\left\langle{x}{y}^{{2}}+{8}{x},{x}^{{2}}{y}-{8}{y}\right\rangle}$$. Compute the flux $$\displaystyle\oint{F}\cdot{n}{d}{s}$$ of F across a simple closed curve that is the boundary of the half-disk given by $$\displaystyle{x}^{{2}}+{y}^{{2}}\le{7},{y}\ge{0}$$ using the vector form Green's Theorem.
$$\displaystyle\oint{F}\cdot{n}{d}{s}=$$ ?