green's theorem

Verify Green’s theorem for the vector field𝐹=(𝑥2−𝑦3)𝑖+(𝑥3+𝑦2)𝑗, over the ellipse 𝐶:𝑥2+4𝑦2=64

[Rik]

Original question:

green's theorem

Verify Green’s theorem for the vector field𝐹=(𝑥2−𝑦3)𝑖+(𝑥3+𝑦2)𝑗, over the ellipse 𝐶:𝑥2+4𝑦2=64

[Rena Berman]

(Answers Dev) Restored edit

[Rik]

@Sanjana Chhabra Why did you edit away your question again? That is very rude to Mehul, who was kind enough to spend time answering your question. Now you repay that kinding by making sure nobody else can benefit from that help.

Is this a homework assignment and are you afraid your instructor will find this question and accuse you of plagiarism?

[Rena Berman]

(Answers Dev) Restored edit

[Gayathri]

integration [(xy+y^2)dx+(x^2)dy]

[Walter Roberson]

@Gayathri I do not understand how that will help prove Green's theorem?

I am also unclear as to which variable the integration is with respect to?

Answer 1

[Mehul Mathur]

clear
clc
syms x y t
F=input('Enter the vector function M(x,y)i+N(x,y)j in the form [M N]: ');
M(x,y)=F(1); N(x,y)=F(2);
r=input('Enter the parametric form of the curve C as [r1(t) r2(t)]: ');
r1=r(1);r2=r(2);
P=M(r1,r2);Q=N(r1,r2);
dr=diff(r,t);
F1=sum([P,Q].*dr);
T=input('Enter the limits of integration for t [t1,t2]: ');
t1=T(1);t2=T(2);
LHS=int(F1,t,t1,t2);
yL=input('Enter limits for y in terms of x: [y1,y2]: ');
xL=input('Enter limits for x as constants: [x1,x2]: ');
y1=yL(1);y2=yL(2);x1=xL(1);x2=xL(2);
F2=diff(N,x)-diff(M,y);
RHS=int(int(F2,y,y1,y2),x,x1,x2);
if(LHS==RHS)
 disp('LHS of Greens theorem=')
 disp(LHS)
 disp('RHS of Greens theorem=')
 disp(RHS)
 disp('Hence Greens theorem is verified.');
end