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 oint_C F*n ds = int int_D div F(x,y) dA where n(t) is the outward unit normal vector to C.

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.

Answers (1)

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
\(\displaystyle\oint_{{C}}{F}.{n}{d}{s}=\int\int_{{D}}{D}{i}{v}{\left({F}\right)}{d}{A}\)
0
 
Best answer

expert advice

Have a similar question?
We can deal with it in 3 hours

Relevant Questions

asked 2021-01-05
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).
asked 2020-10-27
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).
asked 2021-01-19
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.
asked 2021-03-04

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 \(g \times n = Dng\) 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.)
\(\displaystyle\oint_{{c}}{F}\cdot{n}{d}{s}=\int\int_{{D}}\div{F}{\left({x},{y}\right)}{d}{A}\)

...