use green therom to find counter clockwise circulation and outward flux

use green therom to find counter clockwise circulation and outward flux for the field F=(7x-3y)i+(7y-3x)j and the curve C,the square bounded by x=0,x=7,y=0,y=7

i have solved this the flux is 686 and circulation is 0 my code isnt working though incorrect number or types of inputs or outputs for function

% Define the vector field F
syms x y
F = [7*x-3*y, 7*y-3*x];

% Define the integrand as a symbolic expression
G = diff(F(2), x) - diff(F(1), y);

% Convert the integrand to a numeric function
Gfun = matlabFunction(double(G));

% Evaluate the double integral of the integrand over the region D
flux = integral2(Gfun, 0, 7, 0, 7);

% Evaluate the line integral of F around the square
circulation = integral(F(1), 0, 7) + integral(F(2), 0, 7)...
            + integral(-F(1), 7, 0) + integral(-F(2), 7, 0);

% Print the results
disp(['The outward flux of F across the square is ', num2str(flux)]);
disp(['The circulation of F around the square is ', num2str(circulation)]);

Answer 1

[VBBV]

% Define the vector field F
syms x y
F = [7*x-3*y, 7*y-3*x];

% Define the integrand as a symbolic expression for Circulation 
G = diff(F(2), x) - diff(F(1), y)

G = 

0

% Convert the integrand to a numeric function
% Evaluate the double integral of the integrand over the region D
flux = int(int(F, 0, 7), 0, 7)

flux = 

% Evaluate the line integral of F around the square  (counterclockwise
% direction) 
circulation = int(F(1), 0, 7) + int(F(2), 0, 7)...
            + int(-F(1), 7, 0) + int(-F(2),7, 0)

circulation = 

% Print the results
disp(['The outward flux of F across the square is ', num2str(double(flux))]);

The outward flux of F across the square is 686  686

disp(['The circulation of F around the square is ' ,char(circulation)]);

The circulation of F around the square is 56*y + 196

[VBBV]

Use int in place of integral, for finding the circulation. You seem to integrate Gfun defined for vorticity term to find Flux, instead you may need the Function defined for the curve C

[Joshua Thompson]

very nice it gave the flux but gave the circulation as a equation of y

[VBBV]

% Define the vector field F
syms x y
F = [7*x-3*y, 7*y-3*x];

% Define the integrand as a symbolic expression for Circulation 
G = diff(F(2), x) - diff(F(1), y)

G = 

0

% Convert the integrand to a numeric function
% Evaluate the double integral of the integrand over the region D
flux = int(int(F, 0, 7), 0, 7)

flux = 

% Evaluate the line (double) integral of F around the square  (counterclockwise
% direction) 
circulation = int(int(F(1), 0, 7),0,0) + int(int(F(2), 0, 7),7,7)...
            + int(int(-F(1), 7, 0),7,7) + int(int(-F(2),7, 0),0,0)

circulation = 

0

% Print the results
disp(['The outward flux of F across the square is ', num2str(double(flux))]);

The outward flux of F across the square is 686  686

disp(['The circulation of F around the square is ' ,char(circulation)]);

The circulation of F around the square is 0

Answer 2

[Paul]

Isn't the answer 0? Using the approach referenced here

syms x y real
% vector field
F = [7*x-3*y, 7*y-3*x];

% Define the integrand as a symbolic expression
G = diff(F(2), x) - diff(F(1), y)

G = 

0

The area integral must be zero, but compute it anyway

flux = int(int(G,x,0,7),y,0,7);

% Evaluate the line integral of F around the square

% C1 is the segment from (0,0) to (7,0)
% C1 as parmeterized by t: c(t) = (t,0) 0 <= t <= 7
syms t real
c(t) = [t, 0];
dc(t) = diff(c(t),t);
iC1 = int(dot(subs(F,[x y],c(t)),dc(t)),t,0,7);

% C2 is the segment from (7,0) to (7,7)
% C2 as parmeterized by t: c(t) = (7,t) 0 <= t <= 7
c(t) = [7, t];
dc(t) = diff(c(t),t);
dc(t) = diff(c(t),t);
iC2 = int(dot(subs(F,[x y],c(t)),dc(t)),t,0,7);

% C3 is the segment from (7,7) to (0,7)
% C3 as parmeterized by t: c(t) = (t,7) 7 >= t >= 0
c(t) = [t, 7];
dc(t) = diff(c(t),t);
iC3 = int(dot(subs(F,[x y],c(t)),dc(t)),t,7,0);

% C4 is the segment from (0,7) to (0,9)
% C4 as parmeterized by t: c(t) = (0,t) 7 >= t >= 0
c(t) = [0, t];
dc(t) = diff(c(t),t);
dc(t) = diff(c(t),t);
iC4 = int(dot(subs(F,[x y],c(t)),dc(t)),t,7,0);

circulation = iC1 + iC2 + iC3 + iC4;

% Print the results
disp(['The outward flux of F across the square is ', num2str(double(flux))]);

The outward flux of F across the square is 0

disp(['The circulation of F around the square is ', num2str(double(circulation))]);

The circulation of F around the square is 0