Question

Verify Green’s theorem in the plane for ∫c{(x-2y)dx +xdy } taken around the circle x2+y2 =1

Answer 1

Answer:

Basic Algebra Identities

(a + b)2 = a2 + b2 + 2ab

(a – b)2 = a2 + b2 – 2ab

a2 – b2 = (a + b)(a – b)

a2 + b2 = (a + b)2 – 2ab = (a – b)2 + 2ab

a3 + b3 = (a + b)(a2 – ab + b2)

a3 – b3 = (a – b)(a2 + ab + b2)

(a + b)3 = a3 + 3ab(a + b) + b3

(a – b)3 = a3 – 3ab(a – b) – b3

Answer 2

Answer:

Green's theorem does not hold.

Step-by-step explanation:

Green's theorem: The line integral around a closed path enclosing an area equals the surface integral and calculated as

$\int\limits_c {Mdx+Ndy} =\int \int\limits_D \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right)dA$

Given: The enclosed area is around a circle, $x^2+y^2=1$.

Thus, area = $\pi r^2$

                 = $\pi (1)^2$ (Since $r=1$)

                 = $\pi$ unit square.

Consider the given integral as follows:

$\int\limits_c {(x-2y)dx+xdy}$

Using Green's theorem,

$\int\limits_c {(x-2y)dx+xdy} =\int \int\limits_D \left( \frac{\partial }{\partial x}(x) - \frac{\partial }{\partial y}(x-2y) \right)dxdy$

                              $=\int \int\limits_D (1 +2)dxdy$

                              $=3\int \int\limits_D dxdy$  ...... (1)

Consider parametric equations.

$x=acos \theta$

$y=asin \theta$, where $0\leq a\leq 1$ and $0\leq \theta \leq 2\pi$

Then Jacobian is,

$|J(a, \theta)|=\left[\begin{array}{ccc}\frac{\partial x}{\partial a} &\frac{\partial x}{\partial \theta}\\\frac{\partial y}{\partial a} &\frac{\partial y}{\partial \theta} \\ \end{array}\right]$

       $|J|=\left[\begin{array}{ccc} cos \theta &-asin \theta\\sin \theta &acos \theta \\ \end{array}\right]$

            $=a$

Using parametric equation, the integral (1) reduces to

$=3\int \limit^{1}_0 \int\limits^{2\pi}_0 ad\theta da$

$=3\int \limit^{1}_0 a\left(\theta \right)^{2\pi}_0$

$=6\pi\int \limit^{1}_0 a da\\=6\pi \left(\frac{a^2}{2} \right)^1_0\\=3\pi$

Thus, in the plane for the $\int\limits_c {(x-2y)dx+xdy}$ taken around the circle $x^2+y^2=1$, Green's theorem does not hold.

To verify the green's theorem radius of given circle must be $\sqrt{3}$.

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