Green’s Theorem

 Greens theorem

$$M(x,y), N(x,y)\text{ be continuous in a region} R\text{ of  }xy\text{ plane bounded by a closed curve }c\text{ then}$$ $$\oint Mdx+Ndy=\int \int_R \frac{\partial N}{\partial x}-\frac{\partial M}{\partial y} dxdy$$

Algorithm

$$\text{Create a function for LHS: Evaluate }\oint Mdx+Ndy\text{ along the curve using Line integral}$$

$$\text{Create a function for RHS: Evaluate  }\int \int_R \frac{\partial N}{\partial x}-\frac{\partial M}{\partial y} dxdy\text{ using multiple integrals}$$

$$\text{Check whether LHS=RHS or not}$$

%%%%%%%%%%%%%%%% Main Program %%%%%%%%%%%%%%%%%%

clc

clear

 

syms x y

 

LHS=0;

%Get the number of sub path in the curve C

n=input("How many sub curves are ther in the curve C? :")

%Calculate the line integral along each path (LHS)

for i=1:n

    line(i)=LineIntegral(x,y);

    LHS=vpa(LHS+line(i))

end

%Calculate the double integral (RHS)

RHS=vpa(MultiIntegral(x,y))

 

%%%%%%%%%%%%%%%% Main Program ends %%%%%%%%%%%%%%%%%%

%Define function for Line integral

function line = LineIntegral(x,y)

option=input("1: y=f(x)\n2: x=g(y)\n")

 

switch option

    case 1

        y=input("Enter the function f(x): \n")

        r=[x y];

        F=input("Enter the vector function F(x,y)=[F1 F2]: F(x,y)=")

        dr=diff(r,x);

        integrand1=dot(F,dr);

        xL=input("Enter the lower limit of x:");

        xU=input("Enter the Upper limit of x:");

        line=int(integrand1,x,xL,xU)

    case 2

        x=input("Enter the function g(y): \n")

        r=[x y];

        F=input("Enter the vector function F(x,y)=[F1 F2]: F(x,y)=")

        dr=diff(r,y);

        integrand1=dot(F,dr);

        yL=input("Enter the lower limit of y:");

        yU=input("Enter the Upper limit of y:");

        line=int(integrand1,y,yL,yU)

    otherwise

        fprintf("Give the proper input")

end

 

end

 

%%%%%%%%%%%%%%%% Line Integral definition ends %%%%%%%%%%%%%%%%%%

%Define function for Double integral

function d = MultiIntegral(x,y)

 

M= input("Enter the M(x,y): ");

N= input("Enter the N(x,y): ");

f=diff(N,x)-diff(M,y);

 

disp('f(x,y) :');

disp(f);

 

xL=input("Enter lower limit of x: ");

xU=input("Enter Upper limit of x: ");

yL=input("Enter lower limit of y: ");

yU=input("Enter Upper limit of y: ");

 

d = int(int(f,y,yL,yU),x,xL,xU);

end

%%%%%%%%%%%%%%%% Double Integral definition ends %%%%%%%%%%%%%%%%%%

 Example 1

Verify Green’s theorem for $$\int_c (y-\sin(x))dx+cos(y) dy $$ where c is the plane triangle enclosed by the lines $$y=0, x=\pi/2, y=2x/\pi.$$

Output

How many subcurves are ther in the curve C? :3

n =

     3

1: y=f(x)

2: x=g(y)

1

option =

     1

Enter the function f(x):

0

y =

     0

Enter the vector function F(x,y)=[F1 F2]: F(x,y)=[y-sin(x) cos(y)]

F =

[-sin(x), 1]

Enter the lower limit of x:0

Enter the Upper limit of x:pi/2

line =

-1

 

LHS =

-1.0

 

1: y=f(x)

2: x=g(y)

2

option =

     2

Enter the function g(y):

pi/2

x =

    1.5708

Enter the vector function F(x,y)=[F1 F2]: F(x,y)=[y-sin(x) cos(y)]

F =

[- 1 + y, cos(y)]

Enter the lower limit of y:0

Enter the Upper limit of y:1

line =

sin(1)

 

LHS =

-0.1585290151921034933474976783697

 

1: y=f(x)

2: x=g(y)

1

option =

     1

Enter the function f(x):

2*x/pi

y =

(2*x)/pi

Enter the vector function F(x,y)=[F1 F2]: F(x,y)=[y-sin(x) cos(y)]

F =

 [(2*x)/pi - sin(x), cos((2*x)/pi)]

Enter the lower limit of x:pi/2

Enter the Upper limit of x:0

line =

1 - sin(1) - pi/4

 

LHS =

-0.78539816339744830961566084581988

 

Enter the M(x,y): y-sin(x)

Enter the N(x,y): cos(y)

f(x,y) :

-1

 

Enter lower limit of x: 0

Enter Upper limit of x: pi/2

Enter lower limit of y: 0

Enter Upper limit of y: 2*x/pi

 

RHS =

 

-0.78539816339744830961566084581988