green's theorem

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Sanjana Chhabra
Sanjana Chhabra 2022-1-9
评论： Walter Roberson about 3 hours 前
Verify Green’s theorem for the vector field𝐹=(𝑥2−𝑦3)𝑖+(𝑥3+𝑦2)𝑗, over the ellipse 𝐶:𝑥2+4𝑦2=64
  6 个评论
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Gayathri
Gayathri 2023-7-8
integration [(xy+y^2)dx+(x^2)dy]
Walter Roberson
Walter Roberson 2023-7-8
@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?

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回答（2 个）

Mehul Mathur
Mehul Mathur 2022-1-11
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
Charles
Charles about 3 hours 前
Ran in:
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]: ');
Error using input
Support for user input is required, which is not available on this platform.
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

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